Theorem bj-sbeq 32088

Description: Distribute proper substitution through an equality relation. (See sbceqg 3936). (Contributed by BJ, 6-Oct-2018.)

Assertion

Ref	Expression
bj-sbeq	⊢ ([𝑦 / 𝑥]𝐴 = 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑦 / 𝑥⦌𝐵)

Proof of Theorem bj-sbeq

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfcleq 2604	. . . . 5 ⊢ (𝐴 = 𝐵 ↔ ∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵))
2	1	sbbii 1874	. . . 4 ⊢ ([𝑦 / 𝑥]𝐴 = 𝐵 ↔ [𝑦 / 𝑥]∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵))
3		sbsbc 3406	. . . 4 ⊢ ([𝑦 / 𝑥]∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵) ↔ [𝑦 / 𝑥]∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵))
4		sbcal 3452	. . . 4 ⊢ ([𝑦 / 𝑥]∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵) ↔ ∀𝑧[𝑦 / 𝑥](𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵))
5	2, 3, 4	3bitri 285	. . 3 ⊢ ([𝑦 / 𝑥]𝐴 = 𝐵 ↔ ∀𝑧[𝑦 / 𝑥](𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵))
6		vex 3176	. . . . 5 ⊢ 𝑦 ∈ V
7		sbcbig 3447	. . . . 5 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑥](𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵) ↔ ([𝑦 / 𝑥]𝑧 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑧 ∈ 𝐵)))
8	6, 7	ax-mp 5	. . . 4 ⊢ ([𝑦 / 𝑥](𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵) ↔ ([𝑦 / 𝑥]𝑧 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑧 ∈ 𝐵))
9	8	albii 1737	. . 3 ⊢ (∀𝑧[𝑦 / 𝑥](𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵) ↔ ∀𝑧([𝑦 / 𝑥]𝑧 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑧 ∈ 𝐵))
10		sbcel2 3941	. . . . 5 ⊢ ([𝑦 / 𝑥]𝑧 ∈ 𝐴 ↔ 𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐴)
11		sbcel2 3941	. . . . 5 ⊢ ([𝑦 / 𝑥]𝑧 ∈ 𝐵 ↔ 𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵)
12	10, 11	bibi12i 328	. . . 4 ⊢ (([𝑦 / 𝑥]𝑧 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑧 ∈ 𝐵) ↔ (𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐴 ↔ 𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵))
13	12	albii 1737	. . 3 ⊢ (∀𝑧([𝑦 / 𝑥]𝑧 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑧 ∈ 𝐵) ↔ ∀𝑧(𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐴 ↔ 𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵))
14	5, 9, 13	3bitri 285	. 2 ⊢ ([𝑦 / 𝑥]𝐴 = 𝐵 ↔ ∀𝑧(𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐴 ↔ 𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵))
15		dfcleq 2604	. 2 ⊢ (⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑦 / 𝑥⦌𝐵 ↔ ∀𝑧(𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐴 ↔ 𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵))
16	14, 15	bitr4i 266	1 ⊢ ([𝑦 / 𝑥]𝐴 = 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑦 / 𝑥⦌𝐵)

Syntax hints:   ↔ wb 195  ∀wal 1473   = wceq 1475  [wsb 1867   ∈ wcel 1977  Vcvv 3173  [wsbc 3402  ⦋csb 3499

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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-nul 3875

This theorem is referenced by: (None)