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Theorem frege117 43441
Description: Lemma for frege118 43442. Proposition 117 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege116.x 𝑋𝑈
Assertion
Ref Expression
frege117 ((∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)) → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋))) → (Fun 𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋))))
Distinct variable groups:   𝑎,𝑏,𝑅   𝑋,𝑎,𝑏
Allowed substitution hints:   𝑈(𝑎,𝑏)   𝑌(𝑎,𝑏)

Proof of Theorem frege117
StepHypRef Expression
1 frege116.x . . 3 𝑋𝑈
21frege116 43440 . 2 (Fun 𝑅 → ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
3 frege9 43273 . 2 ((Fun 𝑅 → ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋))) → ((∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)) → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋))) → (Fun 𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋)))))
42, 3ax-mp 5 1 ((∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)) → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋))) → (Fun 𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1531   = wceq 1533  wcel 2098   class class class wbr 5152  ccnv 5681  Fun wfun 6547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-frege1 43251  ax-frege2 43252  ax-frege8 43270  ax-frege52a 43318  ax-frege58b 43362
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ifp 1061  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-fun 6555
This theorem is referenced by:  frege118  43442
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