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Theorem frege116 41133
Description: One direction of dffrege115 41132. 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 41132 . 2 (∀𝑐𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ Fun 𝑅)
2 frege116.x . . . 4 𝑋𝑈
32frege68c 41085 . . 3 ((∀𝑐𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ Fun 𝑅) → (Fun 𝑅[𝑋 / 𝑐]𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐))))
4 sbcal 3742 . . . 4 ([𝑋 / 𝑐]𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑏[𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)))
5 sbcimg 3729 . . . . . . 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 5088 . . . . . . . . 9 (𝑋𝑈 → ([𝑋 / 𝑐]𝑏𝑅𝑐𝑏𝑅𝑋 / 𝑐𝑐))
82, 7ax-mp 5 . . . . . . . 8 ([𝑋 / 𝑐]𝑏𝑅𝑐𝑏𝑅𝑋 / 𝑐𝑐)
9 csbvarg 4321 . . . . . . . . . 10 (𝑋𝑈𝑋 / 𝑐𝑐 = 𝑋)
102, 9ax-mp 5 . . . . . . . . 9 𝑋 / 𝑐𝑐 = 𝑋
1110breq2i 5038 . . . . . . . 8 (𝑏𝑅𝑋 / 𝑐𝑐𝑏𝑅𝑋)
128, 11bitri 278 . . . . . . 7 ([𝑋 / 𝑐]𝑏𝑅𝑐𝑏𝑅𝑋)
13 sbcal 3742 . . . . . . . 8 ([𝑋 / 𝑐]𝑎(𝑏𝑅𝑎𝑎 = 𝑐) ↔ ∀𝑎[𝑋 / 𝑐](𝑏𝑅𝑎𝑎 = 𝑐))
14 sbcimg 3729 . . . . . . . . . . 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 3755 . . . . . . . . . . . 12 (𝑋𝑈 → ([𝑋 / 𝑐]𝑏𝑅𝑎𝑏𝑅𝑎))
172, 16ax-mp 5 . . . . . . . . . . 11 ([𝑋 / 𝑐]𝑏𝑅𝑎𝑏𝑅𝑎)
18 sbceq2g 4306 . . . . . . . . . . . . 13 (𝑋𝑈 → ([𝑋 / 𝑐]𝑎 = 𝑐𝑎 = 𝑋 / 𝑐𝑐))
192, 18ax-mp 5 . . . . . . . . . . . 12 ([𝑋 / 𝑐]𝑎 = 𝑐𝑎 = 𝑋 / 𝑐𝑐)
2010eqeq2i 2751 . . . . . . . . . . . 12 (𝑎 = 𝑋 / 𝑐𝑐𝑎 = 𝑋)
2119, 20bitri 278 . . . . . . . . . . 11 ([𝑋 / 𝑐]𝑎 = 𝑐𝑎 = 𝑋)
2217, 21imbi12i 354 . . . . . . . . . 10 (([𝑋 / 𝑐]𝑏𝑅𝑎[𝑋 / 𝑐]𝑎 = 𝑐) ↔ (𝑏𝑅𝑎𝑎 = 𝑋))
2315, 22bitri 278 . . . . . . . . 9 ([𝑋 / 𝑐](𝑏𝑅𝑎𝑎 = 𝑐) ↔ (𝑏𝑅𝑎𝑎 = 𝑋))
2423albii 1826 . . . . . . . 8 (∀𝑎[𝑋 / 𝑐](𝑏𝑅𝑎𝑎 = 𝑐) ↔ ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋))
2513, 24bitri 278 . . . . . . 7 ([𝑋 / 𝑐]𝑎(𝑏𝑅𝑎𝑎 = 𝑐) ↔ ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋))
2612, 25imbi12i 354 . . . . . 6 (([𝑋 / 𝑐]𝑏𝑅𝑐[𝑋 / 𝑐]𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ (𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
276, 26bitri 278 . . . . 5 ([𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ (𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
2827albii 1826 . . . 4 (∀𝑏[𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
294, 28bitri 278 . . 3 ([𝑋 / 𝑐]𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
303, 29syl6ib 254 . 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 209  wal 1540   = wceq 1542  wcel 2114  [wsbc 3680  csb 3790   class class class wbr 5030  ccnv 5524  Fun wfun 6333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-frege1 40944  ax-frege2 40945  ax-frege8 40963  ax-frege52a 41011  ax-frege58b 41055
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-ifp 1063  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-fun 6341
This theorem is referenced by:  frege117  41134
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