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Theorem frege116 37293
 Description: One direction of dffrege115 37292. Proposition 116 of [Frege1879] p. 77. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege116.x 𝑋𝑈
Assertion
Ref Expression
frege116 (Fun 𝑅 → ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
Distinct variable groups:   𝑎,𝑏,𝑅   𝑋,𝑎,𝑏
Allowed substitution hints:   𝑈(𝑎,𝑏)

Proof of Theorem frege116
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 dffrege115 37292 . 2 (∀𝑐𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ Fun 𝑅)
2 frege116.x . . . 4 𝑋𝑈
32frege68c 37245 . . 3 ((∀𝑐𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ Fun 𝑅) → (Fun 𝑅[𝑋 / 𝑐]𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐))))
4 sbcal 3452 . . . 4 ([𝑋 / 𝑐]𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑏[𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)))
5 sbcimg 3444 . . . . . . 7 (𝑋𝑈 → ([𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ([𝑋 / 𝑐]𝑏𝑅𝑐[𝑋 / 𝑐]𝑎(𝑏𝑅𝑎𝑎 = 𝑐))))
62, 5ax-mp 5 . . . . . 6 ([𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ([𝑋 / 𝑐]𝑏𝑅𝑐[𝑋 / 𝑐]𝑎(𝑏𝑅𝑎𝑎 = 𝑐)))
7 sbcbr2g 4640 . . . . . . . . 9 (𝑋𝑈 → ([𝑋 / 𝑐]𝑏𝑅𝑐𝑏𝑅𝑋 / 𝑐𝑐))
82, 7ax-mp 5 . . . . . . . 8 ([𝑋 / 𝑐]𝑏𝑅𝑐𝑏𝑅𝑋 / 𝑐𝑐)
9 csbvarg 3955 . . . . . . . . . 10 (𝑋𝑈𝑋 / 𝑐𝑐 = 𝑋)
102, 9ax-mp 5 . . . . . . . . 9 𝑋 / 𝑐𝑐 = 𝑋
1110breq2i 4591 . . . . . . . 8 (𝑏𝑅𝑋 / 𝑐𝑐𝑏𝑅𝑋)
128, 11bitri 263 . . . . . . 7 ([𝑋 / 𝑐]𝑏𝑅𝑐𝑏𝑅𝑋)
13 sbcal 3452 . . . . . . . 8 ([𝑋 / 𝑐]𝑎(𝑏𝑅𝑎𝑎 = 𝑐) ↔ ∀𝑎[𝑋 / 𝑐](𝑏𝑅𝑎𝑎 = 𝑐))
14 sbcimg 3444 . . . . . . . . . . 11 (𝑋𝑈 → ([𝑋 / 𝑐](𝑏𝑅𝑎𝑎 = 𝑐) ↔ ([𝑋 / 𝑐]𝑏𝑅𝑎[𝑋 / 𝑐]𝑎 = 𝑐)))
152, 14ax-mp 5 . . . . . . . . . 10 ([𝑋 / 𝑐](𝑏𝑅𝑎𝑎 = 𝑐) ↔ ([𝑋 / 𝑐]𝑏𝑅𝑎[𝑋 / 𝑐]𝑎 = 𝑐))
16 sbcg 3470 . . . . . . . . . . . 12 (𝑋𝑈 → ([𝑋 / 𝑐]𝑏𝑅𝑎𝑏𝑅𝑎))
172, 16ax-mp 5 . . . . . . . . . . 11 ([𝑋 / 𝑐]𝑏𝑅𝑎𝑏𝑅𝑎)
18 sbceq2g 3942 . . . . . . . . . . . . 13 (𝑋𝑈 → ([𝑋 / 𝑐]𝑎 = 𝑐𝑎 = 𝑋 / 𝑐𝑐))
192, 18ax-mp 5 . . . . . . . . . . . 12 ([𝑋 / 𝑐]𝑎 = 𝑐𝑎 = 𝑋 / 𝑐𝑐)
2010eqeq2i 2622 . . . . . . . . . . . 12 (𝑎 = 𝑋 / 𝑐𝑐𝑎 = 𝑋)
2119, 20bitri 263 . . . . . . . . . . 11 ([𝑋 / 𝑐]𝑎 = 𝑐𝑎 = 𝑋)
2217, 21imbi12i 339 . . . . . . . . . 10 (([𝑋 / 𝑐]𝑏𝑅𝑎[𝑋 / 𝑐]𝑎 = 𝑐) ↔ (𝑏𝑅𝑎𝑎 = 𝑋))
2315, 22bitri 263 . . . . . . . . 9 ([𝑋 / 𝑐](𝑏𝑅𝑎𝑎 = 𝑐) ↔ (𝑏𝑅𝑎𝑎 = 𝑋))
2423albii 1737 . . . . . . . 8 (∀𝑎[𝑋 / 𝑐](𝑏𝑅𝑎𝑎 = 𝑐) ↔ ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋))
2513, 24bitri 263 . . . . . . 7 ([𝑋 / 𝑐]𝑎(𝑏𝑅𝑎𝑎 = 𝑐) ↔ ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋))
2612, 25imbi12i 339 . . . . . 6 (([𝑋 / 𝑐]𝑏𝑅𝑐[𝑋 / 𝑐]𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ (𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
276, 26bitri 263 . . . . 5 ([𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ (𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
2827albii 1737 . . . 4 (∀𝑏[𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
294, 28bitri 263 . . 3 ([𝑋 / 𝑐]𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
303, 29syl6ib 240 . 2 ((∀𝑐𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ Fun 𝑅) → (Fun 𝑅 → ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋))))
311, 30ax-mp 5 1 (Fun 𝑅 → ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 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:  frege117  37294
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