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Theorem bj-sbceqgALT 32597
Description: Distribute proper substitution through an equality relation. Alternate proof of sbceqg 3962. (Contributed by BJ, 6-Oct-2018.) Proof modification is discouraged to avoid using sbceqg 3962, but "minimize */except sbceqg" is ok. (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-sbceqgALT (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))

Proof of Theorem bj-sbceqgALT
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfcleq 2615 . . . . . 6 (𝐵 = 𝐶 ↔ ∀𝑦(𝑦𝐵𝑦𝐶))
21sbcth 3437 . . . . 5 (𝐴𝑉[𝐴 / 𝑥](𝐵 = 𝐶 ↔ ∀𝑦(𝑦𝐵𝑦𝐶)))
3 sbcbig 3467 . . . . 5 (𝐴𝑉 → ([𝐴 / 𝑥](𝐵 = 𝐶 ↔ ∀𝑦(𝑦𝐵𝑦𝐶)) ↔ ([𝐴 / 𝑥]𝐵 = 𝐶[𝐴 / 𝑥]𝑦(𝑦𝐵𝑦𝐶))))
42, 3mpbid 222 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶[𝐴 / 𝑥]𝑦(𝑦𝐵𝑦𝐶)))
5 sbcal 3472 . . . 4 ([𝐴 / 𝑥]𝑦(𝑦𝐵𝑦𝐶) ↔ ∀𝑦[𝐴 / 𝑥](𝑦𝐵𝑦𝐶))
64, 5syl6bb 276 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ∀𝑦[𝐴 / 𝑥](𝑦𝐵𝑦𝐶)))
7 sbcbig 3467 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥](𝑦𝐵𝑦𝐶) ↔ ([𝐴 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐶)))
87albidv 1846 . . 3 (𝐴𝑉 → (∀𝑦[𝐴 / 𝑥](𝑦𝐵𝑦𝐶) ↔ ∀𝑦([𝐴 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐶)))
9 sbcel2 3967 . . . . . 6 ([𝐴 / 𝑥]𝑦𝐵𝑦𝐴 / 𝑥𝐵)
109a1i 11 . . . . 5 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐵𝑦𝐴 / 𝑥𝐵))
11 sbcel2 3967 . . . . . 6 ([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶)
1211a1i 11 . . . . 5 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶))
1310, 12bibi12d 335 . . . 4 (𝐴𝑉 → (([𝐴 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐶) ↔ (𝑦𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))
1413albidv 1846 . . 3 (𝐴𝑉 → (∀𝑦([𝐴 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐶) ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))
156, 8, 143bitrd 294 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))
16 dfcleq 2615 . 2 (𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶 ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))
1715, 16syl6bbr 278 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478   = wceq 1480  wcel 1987  [wsbc 3422  csb 3519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-nul 3898
This theorem is referenced by: (None)
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