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Theorem bj-sbceqgALT 34708
Description: Distribute proper substitution through an equality relation. Alternate proof of sbceqg 4296. (Contributed by BJ, 6-Oct-2018.) Proof modification is discouraged to avoid using sbceqg 4296, but the Metamath program "MM-PA> MINIMIZE_WITH * / EXCEPT sbceqg" command 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 2731 . . . . . 6 (𝐵 = 𝐶 ↔ ∀𝑦(𝑦𝐵𝑦𝐶))
21sbcth 3694 . . . . 5 (𝐴𝑉[𝐴 / 𝑥](𝐵 = 𝐶 ↔ ∀𝑦(𝑦𝐵𝑦𝐶)))
3 sbcbig 3730 . . . . 5 (𝐴𝑉 → ([𝐴 / 𝑥](𝐵 = 𝐶 ↔ ∀𝑦(𝑦𝐵𝑦𝐶)) ↔ ([𝐴 / 𝑥]𝐵 = 𝐶[𝐴 / 𝑥]𝑦(𝑦𝐵𝑦𝐶))))
42, 3mpbid 235 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶[𝐴 / 𝑥]𝑦(𝑦𝐵𝑦𝐶)))
5 sbcal 3740 . . . 4 ([𝐴 / 𝑥]𝑦(𝑦𝐵𝑦𝐶) ↔ ∀𝑦[𝐴 / 𝑥](𝑦𝐵𝑦𝐶))
64, 5bitrdi 290 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ∀𝑦[𝐴 / 𝑥](𝑦𝐵𝑦𝐶)))
7 sbcbig 3730 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥](𝑦𝐵𝑦𝐶) ↔ ([𝐴 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐶)))
87albidv 1926 . . 3 (𝐴𝑉 → (∀𝑦[𝐴 / 𝑥](𝑦𝐵𝑦𝐶) ↔ ∀𝑦([𝐴 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐶)))
9 sbcel2 4302 . . . . . 6 ([𝐴 / 𝑥]𝑦𝐵𝑦𝐴 / 𝑥𝐵)
109a1i 11 . . . . 5 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐵𝑦𝐴 / 𝑥𝐵))
11 sbcel2 4302 . . . . . 6 ([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶)
1211a1i 11 . . . . 5 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶))
1310, 12bibi12d 349 . . . 4 (𝐴𝑉 → (([𝐴 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐶) ↔ (𝑦𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))
1413albidv 1926 . . 3 (𝐴𝑉 → (∀𝑦([𝐴 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐶) ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))
156, 8, 143bitrd 308 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))
16 dfcleq 2731 . 2 (𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶 ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))
1715, 16bitr4di 292 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1540   = wceq 1542  wcel 2113  [wsbc 3679  csb 3788
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 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-nul 4210
This theorem is referenced by: (None)
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