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Theorem frege120 37297
 Description: Simplified application of one direction of dffrege115 37292. 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 37235 . . 3 (∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋) → [𝐴 / 𝑎](𝑌𝑅𝑎𝑎 = 𝑋))
3 sbcim1 3449 . . . 4 ([𝐴 / 𝑎](𝑌𝑅𝑎𝑎 = 𝑋) → ([𝐴 / 𝑎]𝑌𝑅𝑎[𝐴 / 𝑎]𝑎 = 𝑋))
4 sbcbr2g 4640 . . . . . 6 (𝐴𝑊 → ([𝐴 / 𝑎]𝑌𝑅𝑎𝑌𝑅𝐴 / 𝑎𝑎))
51, 4ax-mp 5 . . . . 5 ([𝐴 / 𝑎]𝑌𝑅𝑎𝑌𝑅𝐴 / 𝑎𝑎)
6 csbvarg 3955 . . . . . . 7 (𝐴𝑊𝐴 / 𝑎𝑎 = 𝐴)
71, 6ax-mp 5 . . . . . 6 𝐴 / 𝑎𝑎 = 𝐴
87breq2i 4591 . . . . 5 (𝑌𝑅𝐴 / 𝑎𝑎𝑌𝑅𝐴)
95, 8bitri 263 . . . 4 ([𝐴 / 𝑎]𝑌𝑅𝑎𝑌𝑅𝐴)
10 sbceq1g 3940 . . . . . 6 (𝐴𝑊 → ([𝐴 / 𝑎]𝑎 = 𝑋𝐴 / 𝑎𝑎 = 𝑋))
111, 10ax-mp 5 . . . . 5 ([𝐴 / 𝑎]𝑎 = 𝑋𝐴 / 𝑎𝑎 = 𝑋)
127eqeq1i 2615 . . . . 5 (𝐴 / 𝑎𝑎 = 𝑋𝐴 = 𝑋)
1311, 12bitri 263 . . . 4 ([𝐴 / 𝑎]𝑎 = 𝑋𝐴 = 𝑋)
143, 9, 133imtr3g 283 . . 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 37296 . 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 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:  frege121  37298
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