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Theorem frege116 37089
Description: One direction of dffrege115 37088. 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 37088 . 2 (∀𝑐𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ Fun 𝑅)
2 frege116.x . . . 4 𝑋𝑈
32frege68c 37041 . . 3 ((∀𝑐𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ Fun 𝑅) → (Fun 𝑅[𝑋 / 𝑐]𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐))))
4 sbcal 3451 . . . 4 ([𝑋 / 𝑐]𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑏[𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)))
5 sbcimg 3443 . . . . . . 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 4634 . . . . . . . . 9 (𝑋𝑈 → ([𝑋 / 𝑐]𝑏𝑅𝑐𝑏𝑅𝑋 / 𝑐𝑐))
82, 7ax-mp 5 . . . . . . . 8 ([𝑋 / 𝑐]𝑏𝑅𝑐𝑏𝑅𝑋 / 𝑐𝑐)
9 csbvarg 3954 . . . . . . . . . 10 (𝑋𝑈𝑋 / 𝑐𝑐 = 𝑋)
102, 9ax-mp 5 . . . . . . . . 9 𝑋 / 𝑐𝑐 = 𝑋
1110breq2i 4585 . . . . . . . 8 (𝑏𝑅𝑋 / 𝑐𝑐𝑏𝑅𝑋)
128, 11bitri 262 . . . . . . 7 ([𝑋 / 𝑐]𝑏𝑅𝑐𝑏𝑅𝑋)
13 sbcal 3451 . . . . . . . 8 ([𝑋 / 𝑐]𝑎(𝑏𝑅𝑎𝑎 = 𝑐) ↔ ∀𝑎[𝑋 / 𝑐](𝑏𝑅𝑎𝑎 = 𝑐))
14 sbcimg 3443 . . . . . . . . . . 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 3469 . . . . . . . . . . . 12 (𝑋𝑈 → ([𝑋 / 𝑐]𝑏𝑅𝑎𝑏𝑅𝑎))
172, 16ax-mp 5 . . . . . . . . . . 11 ([𝑋 / 𝑐]𝑏𝑅𝑎𝑏𝑅𝑎)
18 sbceq2g 3941 . . . . . . . . . . . . 13 (𝑋𝑈 → ([𝑋 / 𝑐]𝑎 = 𝑐𝑎 = 𝑋 / 𝑐𝑐))
192, 18ax-mp 5 . . . . . . . . . . . 12 ([𝑋 / 𝑐]𝑎 = 𝑐𝑎 = 𝑋 / 𝑐𝑐)
2010eqeq2i 2621 . . . . . . . . . . . 12 (𝑎 = 𝑋 / 𝑐𝑐𝑎 = 𝑋)
2119, 20bitri 262 . . . . . . . . . . 11 ([𝑋 / 𝑐]𝑎 = 𝑐𝑎 = 𝑋)
2217, 21imbi12i 338 . . . . . . . . . 10 (([𝑋 / 𝑐]𝑏𝑅𝑎[𝑋 / 𝑐]𝑎 = 𝑐) ↔ (𝑏𝑅𝑎𝑎 = 𝑋))
2315, 22bitri 262 . . . . . . . . 9 ([𝑋 / 𝑐](𝑏𝑅𝑎𝑎 = 𝑐) ↔ (𝑏𝑅𝑎𝑎 = 𝑋))
2423albii 1736 . . . . . . . 8 (∀𝑎[𝑋 / 𝑐](𝑏𝑅𝑎𝑎 = 𝑐) ↔ ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋))
2513, 24bitri 262 . . . . . . 7 ([𝑋 / 𝑐]𝑎(𝑏𝑅𝑎𝑎 = 𝑐) ↔ ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋))
2612, 25imbi12i 338 . . . . . 6 (([𝑋 / 𝑐]𝑏𝑅𝑐[𝑋 / 𝑐]𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ (𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
276, 26bitri 262 . . . . 5 ([𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ (𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
2827albii 1736 . . . 4 (∀𝑏[𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
294, 28bitri 262 . . 3 ([𝑋 / 𝑐]𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
303, 29syl6ib 239 . 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 194  wal 1472   = wceq 1474  wcel 1976  [wsbc 3401  csb 3498   class class class wbr 4577  ccnv 5027  Fun wfun 5784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828  ax-frege1 36900  ax-frege2 36901  ax-frege8 36919  ax-frege52a 36967  ax-frege58b 37011
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-ifp 1006  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-fun 5792
This theorem is referenced by:  frege117  37090
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