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Theorem frege120 40197
Description: Simplified application of one direction of dffrege115 40192. Proposition 120 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
frege116.x 𝑋𝑈
frege118.y 𝑌𝑉
frege120.a 𝐴𝑊
Assertion
Ref Expression
frege120 (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴𝐴 = 𝑋)))

Proof of Theorem frege120
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 frege120.a . . . 4 𝐴𝑊
21frege58c 40135 . . 3 (∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋) → [𝐴 / 𝑎](𝑌𝑅𝑎𝑎 = 𝑋))
3 sbcim1 3829 . . . 4 ([𝐴 / 𝑎](𝑌𝑅𝑎𝑎 = 𝑋) → ([𝐴 / 𝑎]𝑌𝑅𝑎[𝐴 / 𝑎]𝑎 = 𝑋))
4 sbcbr2g 5121 . . . . . 6 (𝐴𝑊 → ([𝐴 / 𝑎]𝑌𝑅𝑎𝑌𝑅𝐴 / 𝑎𝑎))
51, 4ax-mp 5 . . . . 5 ([𝐴 / 𝑎]𝑌𝑅𝑎𝑌𝑅𝐴 / 𝑎𝑎)
6 csbvarg 4387 . . . . . . 7 (𝐴𝑊𝐴 / 𝑎𝑎 = 𝐴)
71, 6ax-mp 5 . . . . . 6 𝐴 / 𝑎𝑎 = 𝐴
87breq2i 5071 . . . . 5 (𝑌𝑅𝐴 / 𝑎𝑎𝑌𝑅𝐴)
95, 8bitri 276 . . . 4 ([𝐴 / 𝑎]𝑌𝑅𝑎𝑌𝑅𝐴)
10 sbceq1g 4370 . . . . . 6 (𝐴𝑊 → ([𝐴 / 𝑎]𝑎 = 𝑋𝐴 / 𝑎𝑎 = 𝑋))
111, 10ax-mp 5 . . . . 5 ([𝐴 / 𝑎]𝑎 = 𝑋𝐴 / 𝑎𝑎 = 𝑋)
127eqeq1i 2831 . . . . 5 (𝐴 / 𝑎𝑎 = 𝑋𝐴 = 𝑋)
1311, 12bitri 276 . . . 4 ([𝐴 / 𝑎]𝑎 = 𝑋𝐴 = 𝑋)
143, 9, 133imtr3g 296 . . 3 ([𝐴 / 𝑎](𝑌𝑅𝑎𝑎 = 𝑋) → (𝑌𝑅𝐴𝐴 = 𝑋))
152, 14syl 17 . 2 (∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋) → (𝑌𝑅𝐴𝐴 = 𝑋))
16 frege116.x . . 3 𝑋𝑈
17 frege118.y . . 3 𝑌𝑉
1816, 17frege119 40196 . 2 ((∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋) → (𝑌𝑅𝐴𝐴 = 𝑋)) → (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴𝐴 = 𝑋))))
1915, 18ax-mp 5 1 (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴𝐴 = 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wal 1528   = wceq 1530  wcel 2107  [wsbc 3776  csb 3887   class class class wbr 5063  ccnv 5553  Fun wfun 6346
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326  ax-frege1 40004  ax-frege2 40005  ax-frege8 40023  ax-frege52a 40071  ax-frege58b 40115
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-ifp 1057  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-fun 6354
This theorem is referenced by:  frege121  40198
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