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Theorem ax13 203
Description: Axiom of Equality. Axiom scheme C12' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Axiom B8 (p. 75) of system S2 of [Tarski] p. 77.
Hypotheses
Ref Expression
ax13.1 A:α
ax13.2 B:α
ax13.3 C:(α → ∗)
Assertion
Ref Expression
ax13 ⊤⊧[[A = B] ⇒ [(CA) ⇒ (CB)]]

Proof of Theorem ax13
StepHypRef Expression
1 wtru 40 . . . . . 6 ⊤:∗
2 ax13.1 . . . . . . 7 A:α
3 ax13.2 . . . . . . 7 B:α
42, 3weqi 68 . . . . . 6 [A = B]:∗
51, 4wct 44 . . . . 5 (⊤, [A = B]):∗
6 ax13.3 . . . . . 6 C:(α → ∗)
76, 2wc 45 . . . . 5 (CA):∗
85, 7simpr 23 . . . 4 ((⊤, [A = B]), (CA))⊧(CA)
91, 4simpr 23 . . . . . 6 (⊤, [A = B])⊧[A = B]
106, 2, 9ceq2 80 . . . . 5 (⊤, [A = B])⊧[(CA) = (CB)]
1110, 7adantr 50 . . . 4 ((⊤, [A = B]), (CA))⊧[(CA) = (CB)]
128, 11mpbi 72 . . 3 ((⊤, [A = B]), (CA))⊧(CB)
1312ex 148 . 2 (⊤, [A = B])⊧[(CA) ⇒ (CB)]
1413ex 148 1 ⊤⊧[[A = B] ⇒ [(CA) ⇒ (CB)]]
Colors of variables: type var term
Syntax hints:  ht 2  hb 3  kc 5   = ke 7  kt 8  [kbr 9  kct 10  wffMMJ2 11  wffMMJ2t 12  tim 111
This theorem was proved from axioms:  ax-syl 15  ax-jca 17  ax-simpl 20  ax-simpr 21  ax-id 24  ax-trud 26  ax-cb1 29  ax-cb2 30  ax-refl 39  ax-eqmp 42  ax-ded 43  ax-ceq 46  ax-beta 60  ax-distrc 61  ax-leq 62  ax-distrl 63  ax-hbl1 93  ax-17 95  ax-inst 103
This theorem depends on definitions:  df-ov 65  df-an 118  df-im 119
This theorem is referenced by: (None)
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