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Theorem frege117 42721
Description: Lemma for frege118 42722. 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 42720 . 2 (Fun 𝑅 → ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
3 frege9 42553 . 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 1539   = wceq 1541  wcel 2106   class class class wbr 5148  ccnv 5675  Fun wfun 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-frege1 42531  ax-frege2 42532  ax-frege8 42550  ax-frege52a 42598  ax-frege58b 42642
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-fun 6545
This theorem is referenced by:  frege118  42722
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