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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | idhe 37101 | The identity relation is hereditary in any class. (Contributed by RP, 28-Mar-2020.) |
⊢ I hereditary 𝐴 | ||
Theorem | psshepw 37102 | The relation between sets and their proper subsets is hereditary in the powerclass of any class. (Contributed by RP, 28-Mar-2020.) |
⊢ ◡ [⊊] hereditary 𝒫 𝐴 | ||
Theorem | sshepw 37103 | The relation between sets and their subsets is hereditary in the powerclass of any class. (Contributed by RP, 28-Mar-2020.) |
⊢ (◡ [⊊] ∪ I ) hereditary 𝒫 𝐴 | ||
Axiom | ax-frege1 37104 | The case in which 𝜑 is denied, 𝜓 is affirmed, and 𝜑 is affirmed is excluded. This is evident since 𝜑 cannot at the same time be denied and affirmed. Axiom 1 of [Frege1879] p. 26. Identical to ax-1 6. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜑)) | ||
Axiom | ax-frege2 37105 | If a proposition 𝜒 is a necessary consequence of two propositions 𝜓 and 𝜑 and one of those, 𝜓, is in turn a necessary consequence of the other, 𝜑, then the proposition 𝜒 is a necessary consequence of the latter one, 𝜑, alone. Axiom 2 of [Frege1879] p. 26. Identical to ax-2 7. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))) | ||
Theorem | rp-simp2-frege 37106 | Simplification of triple conjunction. Compare with simp2 1055. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜓))) | ||
Theorem | rp-simp2 37107 | Simplification of triple conjunction. Identical to simp2 1055. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜓) | ||
Theorem | rp-frege3g 37108 |
Add antecedent to ax-frege2 37105. More general statement than frege3 37109.
Like ax-frege2 37105, it is essentially a closed form of mpd 15,
however it
has an extra antecedent.
It would be more natural to prove from a1i 11 and ax-frege2 37105 in Metamath. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 → ((𝜓 → (𝜒 → 𝜃)) → ((𝜓 → 𝜒) → (𝜓 → 𝜃)))) | ||
Theorem | frege3 37109 | Add antecedent to ax-frege2 37105. Special case of rp-frege3g 37108. Proposition 3 of [Frege1879] p. 29. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → 𝜓) → ((𝜒 → (𝜑 → 𝜓)) → ((𝜒 → 𝜑) → (𝜒 → 𝜓)))) | ||
Theorem | rp-misc1-frege 37110 | Double-use of ax-frege2 37105. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜓)) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒))) | ||
Theorem | rp-frege24 37111 | Introducing an embedded antecedent. Alternate proof for frege24 37129. Closed form for a1d 25. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜒 → 𝜓))) | ||
Theorem | rp-frege4g 37112 | Deduction related to distribution. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → (𝜑 → ((𝜓 → 𝜒) → (𝜓 → 𝜃)))) | ||
Theorem | frege4 37113 | Special case of closed form of a2d 29. Special case of rp-frege4g 37112. Proposition 4 of [Frege1879] p. 31. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (((𝜑 → 𝜓) → (𝜒 → (𝜑 → 𝜓))) → ((𝜑 → 𝜓) → ((𝜒 → 𝜑) → (𝜒 → 𝜓)))) | ||
Theorem | frege5 37114 | A closed form of syl 17. Identical to imim2 56. Theorem *2.05 of [WhiteheadRussell] p. 100. Proposition 5 of [Frege1879] p. 32. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → 𝜓) → ((𝜒 → 𝜑) → (𝜒 → 𝜓))) | ||
Theorem | rp-7frege 37115 | Distribute antecedent and add another. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜃 → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))) | ||
Theorem | rp-4frege 37116 | Elimination of a nested antecedent of special form. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝜑 → ((𝜓 → 𝜑) → 𝜒)) → (𝜑 → 𝜒)) | ||
Theorem | rp-6frege 37117 | Elimination of a nested antecedent of special form. (Contributed by RP, 24-Dec-2019.) |
⊢ (𝜑 → ((𝜓 → ((𝜒 → 𝜓) → 𝜃)) → (𝜓 → 𝜃))) | ||
Theorem | rp-8frege 37118 | Eliminate antecedent when it is implied by previous antecedent. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝜑 → (𝜓 → ((𝜒 → 𝜓) → 𝜃))) → (𝜑 → (𝜓 → 𝜃))) | ||
Theorem | rp-frege25 37119 | Closed form for a1dd 48. Alternate route to Proposition 25 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → (𝜓 → (𝜃 → 𝜒)))) | ||
Theorem | frege6 37120 | A closed form of imim2d 55 which is a deduction adding nested antecedents. Proposition 6 of [Frege1879] p. 33. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → ((𝜃 → 𝜓) → (𝜃 → 𝜒)))) | ||
Theorem | axfrege8 37121 |
Swap antecedents. Identical to pm2.04 88. This demonstrates that Axiom 8
of [Frege1879] p. 35 is redundant.
Proof follows closely proof of pm2.04 88 in http://us.metamath.org/mmsolitaire/pmproofs.txt, but in the style of Frege's 1879 work. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))) | ||
Theorem | frege7 37122 | A closed form of syl6 34. The first antecedent is used to replace the consequent of the second antecedent. Proposition 7 of [Frege1879] p. 34. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → 𝜓) → ((𝜒 → (𝜃 → 𝜑)) → (𝜒 → (𝜃 → 𝜓)))) | ||
Axiom | ax-frege8 37123 | Swap antecedents. If two conditions have a proposition as a consequence, their order is immaterial. Third axiom of Frege's 1879 work but identical to pm2.04 88 which can be proved from only ax-mp 5, ax-frege1 37104, and ax-frege2 37105. (Redundant) Axiom 8 of [Frege1879] p. 35. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))) | ||
Theorem | frege26 37124 | Identical to idd 24. Proposition 26 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜓)) | ||
Theorem | frege27 37125 | We cannot (at the same time) affirm 𝜑 and deny 𝜑. Identical to id 22. Proposition 27 of [Frege1879] p. 43. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 → 𝜑) | ||
Theorem | frege9 37126 | Closed form of syl 17 with swapped antecedents. This proposition differs from frege5 37114 only in an unessential way. Identical to imim1 81. Proposition 9 of [Frege1879] p. 35. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))) | ||
Theorem | frege12 37127 | A closed form of com23 84. Proposition 12 of [Frege1879] p. 37. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → (𝜑 → (𝜒 → (𝜓 → 𝜃)))) | ||
Theorem | frege11 37128 | Elimination of a nested antecedent as a partial converse of ja 172. If the proposition that 𝜓 takes place or 𝜑 does not is a sufficient condition for 𝜒, then 𝜓 by itself is a sufficient condition for 𝜒. Identical to jarr 104. Proposition 11 of [Frege1879] p. 36. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒)) | ||
Theorem | frege24 37129 | Closed form for a1d 25. Deduction introducing an embedded antecedent. Identical to rp-frege24 37111 which was proved without relying on ax-frege8 37123. Proposition 24 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜒 → 𝜓))) | ||
Theorem | frege16 37130 | A closed form of com34 89. Proposition 16 of [Frege1879] p. 38. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏)))) → (𝜑 → (𝜓 → (𝜃 → (𝜒 → 𝜏))))) | ||
Theorem | frege25 37131 | Closed form for a1dd 48. Proposition 25 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → (𝜓 → (𝜃 → 𝜒)))) | ||
Theorem | frege18 37132 | Closed form of a syllogism followed by a swap of antecedents. Proposition 18 of [Frege1879] p. 39. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜃 → 𝜑) → (𝜓 → (𝜃 → 𝜒)))) | ||
Theorem | frege22 37133 | A closed form of com45 95. Proposition 22 of [Frege1879] p. 41. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))) → (𝜑 → (𝜓 → (𝜒 → (𝜏 → (𝜃 → 𝜂)))))) | ||
Theorem | frege10 37134 | Result commuting antecedents within an antecedent. Proposition 10 of [Frege1879] p. 36. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (((𝜑 → (𝜓 → 𝜒)) → 𝜃) → ((𝜓 → (𝜑 → 𝜒)) → 𝜃)) | ||
Theorem | frege17 37135 | A closed form of com3l 87. Proposition 17 of [Frege1879] p. 39. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → (𝜓 → (𝜒 → (𝜑 → 𝜃)))) | ||
Theorem | frege13 37136 | A closed form of com3r 85. Proposition 13 of [Frege1879] p. 37. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → (𝜒 → (𝜑 → (𝜓 → 𝜃)))) | ||
Theorem | frege14 37137 | Closed form of a deduction based on com3r 85. Proposition 14 of [Frege1879] p. 37. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏)))) → (𝜑 → (𝜃 → (𝜓 → (𝜒 → 𝜏))))) | ||
Theorem | frege19 37138 | A closed form of syl6 34. Proposition 19 of [Frege1879] p. 39. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜒 → 𝜃) → (𝜑 → (𝜓 → 𝜃)))) | ||
Theorem | frege23 37139 | Syllogism followed by rotation of three antecedents. Proposition 23 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → ((𝜏 → 𝜑) → (𝜓 → (𝜒 → (𝜏 → 𝜃))))) | ||
Theorem | frege15 37140 | A closed form of com4r 92. Proposition 15 of [Frege1879] p. 38. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏)))) → (𝜃 → (𝜑 → (𝜓 → (𝜒 → 𝜏))))) | ||
Theorem | frege21 37141 | Replace antecedent in antecedent. Proposition 21 of [Frege1879] p. 40. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (((𝜑 → 𝜓) → 𝜒) → ((𝜑 → 𝜃) → ((𝜃 → 𝜓) → 𝜒))) | ||
Theorem | frege20 37142 | A closed form of syl8 74. Proposition 20 of [Frege1879] p. 40. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → ((𝜃 → 𝜏) → (𝜑 → (𝜓 → (𝜒 → 𝜏))))) | ||
Theorem | axfrege28 37143 | Contraposition. Identical to con3 148. Theorem *2.16 of [WhiteheadRussell] p. 103. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑)) | ||
Axiom | ax-frege28 37144 | Contraposition. Identical to con3 148. Theorem *2.16 of [WhiteheadRussell] p. 103. Axiom 28 of [Frege1879] p. 43. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.) |
⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑)) | ||
Theorem | frege29 37145 | Closed form of con3d 147. Proposition 29 of [Frege1879] p. 43. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → (¬ 𝜒 → ¬ 𝜓))) | ||
Theorem | frege30 37146 | Commuted, closed form of con3d 147. Proposition 30 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (¬ 𝜒 → ¬ 𝜑))) | ||
Theorem | axfrege31 37147 | Identical to notnotr 124. Axiom 31 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) |
⊢ (¬ ¬ 𝜑 → 𝜑) | ||
Axiom | ax-frege31 37148 | 𝜑 cannot be denied and (at the same time ) ¬ ¬ 𝜑 affirmed. Duplex negatio affirmat. The denial of the denial is affirmation. Identical to notnotr 124. Axiom 31 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.) |
⊢ (¬ ¬ 𝜑 → 𝜑) | ||
Theorem | frege32 37149 | Deduce con1 142 from con3 148. Proposition 32 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (((¬ 𝜑 → 𝜓) → (¬ 𝜓 → ¬ ¬ 𝜑)) → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑))) | ||
Theorem | frege33 37150 | If 𝜑 or 𝜓 takes place, then 𝜓 or 𝜑 takes place. Identical to con1 142. Proposition 33 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)) | ||
Theorem | frege34 37151 | If as a conseqence of the occurence of the circumstance 𝜑, when the obstacle 𝜓 is removed, 𝜒 takes place, then from the circumstance that 𝜒 does not take place while 𝜑 occurs the occurence of the obstacle 𝜓 can be inferred. Closed form of con1d 138. Proposition 34 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (¬ 𝜓 → 𝜒)) → (𝜑 → (¬ 𝜒 → 𝜓))) | ||
Theorem | frege35 37152 | Commuted, closed form of con1d 138. Proposition 35 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (¬ 𝜓 → 𝜒)) → (¬ 𝜒 → (𝜑 → 𝜓))) | ||
Theorem | frege36 37153 | The case in which 𝜓 is denied, ¬ 𝜑 is affirmed, and 𝜑 is affirmed does not occur. If 𝜑 occurs, then (at least) one of the two, 𝜑 or 𝜓, takes place (no matter what 𝜓 might be). Identical to pm2.24 120. Proposition 36 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 → (¬ 𝜑 → 𝜓)) | ||
Theorem | frege37 37154 | If 𝜒 is a necessary consequence of the occurrence of 𝜓 or 𝜑, then 𝜒 is a necessary consequence of 𝜑 alone. Similar to a closed form of orcs 408. Proposition 37 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (((¬ 𝜑 → 𝜓) → 𝜒) → (𝜑 → 𝜒)) | ||
Theorem | frege38 37155 | Identical to pm2.21 119. Proposition 38 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (¬ 𝜑 → (𝜑 → 𝜓)) | ||
Theorem | frege39 37156 | Syllogism between pm2.18 121 and pm2.24 120. Proposition 39 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((¬ 𝜑 → 𝜑) → (¬ 𝜑 → 𝜓)) | ||
Theorem | frege40 37157 | Anything implies pm2.18 121. Proposition 40 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (¬ 𝜑 → ((¬ 𝜓 → 𝜓) → 𝜓)) | ||
Theorem | axfrege41 37158 | Identical to notnot 135. Axiom 41 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) |
⊢ (𝜑 → ¬ ¬ 𝜑) | ||
Axiom | ax-frege41 37159 | The affirmation of 𝜑 denies the denial of 𝜑. Identical to notnot 135. Axiom 41 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.) |
⊢ (𝜑 → ¬ ¬ 𝜑) | ||
Theorem | frege42 37160 | Not not id 22. Proposition 42 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ¬ ¬ (𝜑 → 𝜑) | ||
Theorem | frege43 37161 | If there is a choice only between 𝜑 and 𝜑, then 𝜑 takes place. Identical to pm2.18 121. Proposition 43 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((¬ 𝜑 → 𝜑) → 𝜑) | ||
Theorem | frege44 37162 | Similar to a commuted pm2.62 424. Proposition 44 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((¬ 𝜑 → 𝜓) → ((𝜓 → 𝜑) → 𝜑)) | ||
Theorem | frege45 37163 | Deduce pm2.6 181 from con1 142. Proposition 45 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)) → ((¬ 𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓))) | ||
Theorem | frege46 37164 | If 𝜓 holds when 𝜑 occurs as well as when 𝜑 does not occur, then 𝜓 holds. If 𝜓 or 𝜑 occurs and if the occurences of 𝜑 has 𝜓 as a necessary consequence, then 𝜓 takes place. Identical to pm2.6 181. Proposition 46 of [Frege1879] p. 48. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((¬ 𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓)) | ||
Theorem | frege47 37165 | Deduce consequence follows from either path implied by a disjunction. If 𝜑, as well as 𝜓 is sufficient condition for 𝜒 and 𝜓 or 𝜑 takes place, then the proposition 𝜒 holds. Proposition 47 of [Frege1879] p. 48. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((¬ 𝜑 → 𝜓) → ((𝜓 → 𝜒) → ((𝜑 → 𝜒) → 𝜒))) | ||
Theorem | frege48 37166 | Closed form of syllogism with internal disjunction. If 𝜑 is a sufficient condition for the occurence of 𝜒 or 𝜓 and if 𝜒, as well as 𝜓, is a sufficient condition for 𝜃, then 𝜑 is a sufficient condition for 𝜃. See application in frege101 37278. Proposition 48 of [Frege1879] p. 49. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (¬ 𝜓 → 𝜒)) → ((𝜒 → 𝜃) → ((𝜓 → 𝜃) → (𝜑 → 𝜃)))) | ||
Theorem | frege49 37167 | Closed form of deduction with disjunction. Proposition 49 of [Frege1879] p. 49. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((¬ 𝜑 → 𝜓) → ((𝜑 → 𝜒) → ((𝜓 → 𝜒) → 𝜒))) | ||
Theorem | frege50 37168 | Closed form of jaoi 393. Proposition 50 of [Frege1879] p. 49. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → 𝜓) → ((𝜒 → 𝜓) → ((¬ 𝜑 → 𝜒) → 𝜓))) | ||
Theorem | frege51 37169 | Compare with jaod 394. Proposition 51 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜃 → 𝜒) → (𝜑 → ((¬ 𝜓 → 𝜃) → 𝜒)))) | ||
Here we leverage df-ifp 1007 to partition a wff into two that are disjoint with the selector wff. Thus if we are given ⊢ (𝜑 ↔ if-(𝜓, 𝜒, 𝜃)) then we replace the concept (illegal in our notation ) (𝜑‘𝜓) with if-(𝜓, 𝜒, 𝜃) to reason about the values of the "function." Likewise, we replace the similarly illegal concept ∀𝜓𝜑 with (𝜒 ∧ 𝜃). | ||
Theorem | axfrege52a 37170 | Justification for ax-frege52a 37171. (Contributed by RP, 17-Apr-2020.) |
⊢ ((𝜑 ↔ 𝜓) → (if-(𝜑, 𝜃, 𝜒) → if-(𝜓, 𝜃, 𝜒))) | ||
Axiom | ax-frege52a 37171 | The case when the content of 𝜑 is identical with the content of 𝜓 and in which a proposition controlled by an element for which we substitute the content of 𝜑 is affirmed ( in this specific case the identity logical funtion ) and the same proposition, this time where we subsituted the content of 𝜓, is denied does not take place. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.) |
⊢ ((𝜑 ↔ 𝜓) → (if-(𝜑, 𝜃, 𝜒) → if-(𝜓, 𝜃, 𝜒))) | ||
Theorem | frege52aid 37172 | The case when the content of 𝜑 is identical with the content of 𝜓 and in which 𝜑 is affirmed and 𝜓 is denied does not take place. Identical to biimp 204. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓)) | ||
Theorem | frege53aid 37173 | Specialization of frege53a 37174. Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 → ((𝜑 ↔ 𝜓) → 𝜓)) | ||
Theorem | frege53a 37174 | Lemma for frege55a 37182. Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (if-(𝜑, 𝜃, 𝜒) → ((𝜑 ↔ 𝜓) → if-(𝜓, 𝜃, 𝜒))) | ||
Theorem | axfrege54a 37175 | Justification for ax-frege54a 37176. Identical to biid 250. (Contributed by RP, 24-Dec-2019.) |
⊢ (𝜑 ↔ 𝜑) | ||
Axiom | ax-frege54a 37176 | Reflexive equality of wffs. The content of 𝜑 is identical with the content of 𝜑. Part of Axiom 54 of [Frege1879] p. 50. Identical to biid 250. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.) |
⊢ (𝜑 ↔ 𝜑) | ||
Theorem | frege54cor0a 37177 | Synonym for logical equivalence. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜓 ↔ 𝜑) ↔ if-(𝜓, 𝜑, ¬ 𝜑)) | ||
Theorem | frege54cor1a 37178 | Reflexive equality. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ if-(𝜑, 𝜑, ¬ 𝜑) | ||
Theorem | frege55aid 37179 | Lemma for frege57aid 37186. Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑)) | ||
Theorem | frege55lem1a 37180 | Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜏 → if-(𝜓, 𝜑, ¬ 𝜑)) → (𝜏 → (𝜓 ↔ 𝜑))) | ||
Theorem | frege55lem2a 37181 | Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 ↔ 𝜓) → if-(𝜓, 𝜑, ¬ 𝜑)) | ||
Theorem | frege55a 37182 | Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 ↔ 𝜓) → if-(𝜓, 𝜑, ¬ 𝜑)) | ||
Theorem | frege55cor1a 37183 | Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑)) | ||
Theorem | frege56aid 37184 | Lemma for frege57aid 37186. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (((𝜑 ↔ 𝜓) → (𝜑 → 𝜓)) → ((𝜓 ↔ 𝜑) → (𝜑 → 𝜓))) | ||
Theorem | frege56a 37185 | Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (((𝜑 ↔ 𝜓) → (if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃))) → ((𝜓 ↔ 𝜑) → (if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃)))) | ||
Theorem | frege57aid 37186 | This is the all imporant formula which allows us to apply Frege-style definitions and explore their consequences. A closed form of biimpri 217. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑)) | ||
Theorem | frege57a 37187 | Analogue of frege57aid 37186. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 ↔ 𝜓) → (if-(𝜓, 𝜒, 𝜃) → if-(𝜑, 𝜒, 𝜃))) | ||
Theorem | axfrege58a 37188 | Identical to anifp 1014. Justification for ax-frege58a 37189. (Contributed by RP, 28-Mar-2020.) |
⊢ ((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒)) | ||
Axiom | ax-frege58a 37189 | If ∀𝑥𝜑 is affirmed, [𝑦 / 𝑥]𝜑 cannot be denied. Identical to stdpc4 2341. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 28-Mar-2020.) (New usage is discouraged.) |
⊢ ((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒)) | ||
Theorem | frege58acor 37190 | Lemma for frege59a 37191. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.) |
⊢ (((𝜓 → 𝜒) ∧ (𝜃 → 𝜏)) → (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏))) | ||
Theorem | frege59a 37191 |
A kind of Aristotelian inference. Namely Felapton or Fesapo. Proposition
59 of [Frege1879] p. 51.
Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collection From Frege to Goedel, this proof has the frege12 37127 incorrectly referenced where frege30 37146 is in the original. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.) |
⊢ (if-(𝜑, 𝜓, 𝜃) → (¬ if-(𝜑, 𝜒, 𝜏) → ¬ ((𝜓 → 𝜒) ∧ (𝜃 → 𝜏)))) | ||
Theorem | frege60a 37192 | Swap antecedents of ax-frege58a 37189. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.) |
⊢ (((𝜓 → (𝜒 → 𝜃)) ∧ (𝜏 → (𝜂 → 𝜁))) → (if-(𝜑, 𝜒, 𝜂) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁)))) | ||
Theorem | frege61a 37193 | Lemma for frege65a 37197. Proposition 61 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.) |
⊢ ((if-(𝜑, 𝜓, 𝜒) → 𝜃) → ((𝜓 ∧ 𝜒) → 𝜃)) | ||
Theorem | frege62a 37194 | A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2551 when the minor premise has a particular context. Proposition 62 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.) |
⊢ (if-(𝜑, 𝜓, 𝜃) → (((𝜓 → 𝜒) ∧ (𝜃 → 𝜏)) → if-(𝜑, 𝜒, 𝜏))) | ||
Theorem | frege63a 37195 | Proposition 63 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.) |
⊢ (if-(𝜑, 𝜓, 𝜃) → (𝜂 → (((𝜓 → 𝜒) ∧ (𝜃 → 𝜏)) → if-(𝜑, 𝜒, 𝜏)))) | ||
Theorem | frege64a 37196 | Lemma for frege65a 37197. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.) |
⊢ ((if-(𝜑, 𝜓, 𝜏) → if-(𝜎, 𝜒, 𝜂)) → (((𝜒 → 𝜃) ∧ (𝜂 → 𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜎, 𝜃, 𝜁)))) | ||
Theorem | frege65a 37197 | A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2551 when the minor premise has a general context. Proposition 65 of [Frege1879] p. 53. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.) |
⊢ (((𝜓 → 𝜒) ∧ (𝜏 → 𝜂)) → (((𝜒 → 𝜃) ∧ (𝜂 → 𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁)))) | ||
Theorem | frege66a 37198 | Swap antecedents of frege65a 37197. Proposition 66 of [Frege1879] p. 54. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.) |
⊢ (((𝜒 → 𝜃) ∧ (𝜂 → 𝜁)) → (((𝜓 → 𝜒) ∧ (𝜏 → 𝜂)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁)))) | ||
Theorem | frege67a 37199 | Lemma for frege68a 37200. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.) |
⊢ ((((𝜓 ∧ 𝜒) ↔ 𝜃) → (𝜃 → (𝜓 ∧ 𝜒))) → (((𝜓 ∧ 𝜒) ↔ 𝜃) → (𝜃 → if-(𝜑, 𝜓, 𝜒)))) | ||
Theorem | frege68a 37200 | Combination of applying a definition and applying it to a specific instance. Proposition 68 of [Frege1879] p. 54. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.) |
⊢ (((𝜓 ∧ 𝜒) ↔ 𝜃) → (𝜃 → if-(𝜑, 𝜓, 𝜒))) |
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