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Theorem frege118 37295
 Description: Simplified application of one direction of dffrege115 37292. Proposition 118 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
frege116.x 𝑋𝑈
frege118.y 𝑌𝑉
Assertion
Ref Expression
frege118 (Fun 𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋)))
Distinct variable groups:   𝑅,𝑎   𝑋,𝑎   𝑌,𝑎
Allowed substitution hints:   𝑈(𝑎)   𝑉(𝑎)

Proof of Theorem frege118
Dummy variable 𝑏 is distinct from all other variables.
StepHypRef Expression
1 frege118.y . . . 4 𝑌𝑉
21frege58c 37235 . . 3 (∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)) → [𝑌 / 𝑏](𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
3 sbcimg 3444 . . . . 5 (𝑌𝑉 → ([𝑌 / 𝑏](𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)) ↔ ([𝑌 / 𝑏]𝑏𝑅𝑋[𝑌 / 𝑏]𝑎(𝑏𝑅𝑎𝑎 = 𝑋))))
41, 3ax-mp 5 . . . 4 ([𝑌 / 𝑏](𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)) ↔ ([𝑌 / 𝑏]𝑏𝑅𝑋[𝑌 / 𝑏]𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
5 sbcbr1g 4639 . . . . . . 7 (𝑌𝑉 → ([𝑌 / 𝑏]𝑏𝑅𝑋𝑌 / 𝑏𝑏𝑅𝑋))
61, 5ax-mp 5 . . . . . 6 ([𝑌 / 𝑏]𝑏𝑅𝑋𝑌 / 𝑏𝑏𝑅𝑋)
7 csbvarg 3955 . . . . . . . 8 (𝑌𝑉𝑌 / 𝑏𝑏 = 𝑌)
81, 7ax-mp 5 . . . . . . 7 𝑌 / 𝑏𝑏 = 𝑌
98breq1i 4590 . . . . . 6 (𝑌 / 𝑏𝑏𝑅𝑋𝑌𝑅𝑋)
106, 9bitri 263 . . . . 5 ([𝑌 / 𝑏]𝑏𝑅𝑋𝑌𝑅𝑋)
11 sbcal 3452 . . . . . 6 ([𝑌 / 𝑏]𝑎(𝑏𝑅𝑎𝑎 = 𝑋) ↔ ∀𝑎[𝑌 / 𝑏](𝑏𝑅𝑎𝑎 = 𝑋))
12 sbcimg 3444 . . . . . . . . 9 (𝑌𝑉 → ([𝑌 / 𝑏](𝑏𝑅𝑎𝑎 = 𝑋) ↔ ([𝑌 / 𝑏]𝑏𝑅𝑎[𝑌 / 𝑏]𝑎 = 𝑋)))
131, 12ax-mp 5 . . . . . . . 8 ([𝑌 / 𝑏](𝑏𝑅𝑎𝑎 = 𝑋) ↔ ([𝑌 / 𝑏]𝑏𝑅𝑎[𝑌 / 𝑏]𝑎 = 𝑋))
14 sbcbr1g 4639 . . . . . . . . . . 11 (𝑌𝑉 → ([𝑌 / 𝑏]𝑏𝑅𝑎𝑌 / 𝑏𝑏𝑅𝑎))
151, 14ax-mp 5 . . . . . . . . . 10 ([𝑌 / 𝑏]𝑏𝑅𝑎𝑌 / 𝑏𝑏𝑅𝑎)
168breq1i 4590 . . . . . . . . . 10 (𝑌 / 𝑏𝑏𝑅𝑎𝑌𝑅𝑎)
1715, 16bitri 263 . . . . . . . . 9 ([𝑌 / 𝑏]𝑏𝑅𝑎𝑌𝑅𝑎)
18 sbcg 3470 . . . . . . . . . 10 (𝑌𝑉 → ([𝑌 / 𝑏]𝑎 = 𝑋𝑎 = 𝑋))
191, 18ax-mp 5 . . . . . . . . 9 ([𝑌 / 𝑏]𝑎 = 𝑋𝑎 = 𝑋)
2017, 19imbi12i 339 . . . . . . . 8 (([𝑌 / 𝑏]𝑏𝑅𝑎[𝑌 / 𝑏]𝑎 = 𝑋) ↔ (𝑌𝑅𝑎𝑎 = 𝑋))
2113, 20bitri 263 . . . . . . 7 ([𝑌 / 𝑏](𝑏𝑅𝑎𝑎 = 𝑋) ↔ (𝑌𝑅𝑎𝑎 = 𝑋))
2221albii 1737 . . . . . 6 (∀𝑎[𝑌 / 𝑏](𝑏𝑅𝑎𝑎 = 𝑋) ↔ ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋))
2311, 22bitri 263 . . . . 5 ([𝑌 / 𝑏]𝑎(𝑏𝑅𝑎𝑎 = 𝑋) ↔ ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋))
2410, 23imbi12i 339 . . . 4 (([𝑌 / 𝑏]𝑏𝑅𝑋[𝑌 / 𝑏]𝑎(𝑏𝑅𝑎𝑎 = 𝑋)) ↔ (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋)))
254, 24bitri 263 . . 3 ([𝑌 / 𝑏](𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)) ↔ (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋)))
262, 25sylib 207 . 2 (∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)) → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋)))
27 frege116.x . . 3 𝑋𝑈
2827frege117 37294 . 2 ((∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)) → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋))) → (Fun 𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋))))
2926, 28ax-mp 5 1 (Fun 𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   = wceq 1475   ∈ wcel 1977  [wsbc 3402  ⦋csb 3499   class class class wbr 4583  ◡ccnv 5037  Fun wfun 5798 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-frege1 37104  ax-frege2 37105  ax-frege8 37123  ax-frege52a 37171  ax-frege58b 37215 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-fun 5806 This theorem is referenced by:  frege119  37296
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