Theorem bj-sbceqgALT 32089

Description: Distribute proper substitution through an equality relation. Alternate proof of sbceqg 3936. (Contributed by BJ, 6-Oct-2018.) Proof modification is discouraged to avoid using sbceqg 3936, but "minimize */except sbceqg" is ok. (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-sbceqgALT	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶))

Proof of Theorem bj-sbceqgALT

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfcleq 2604	. . . . . 6 ⊢ (𝐵 = 𝐶 ↔ ∀𝑦(𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶))
2	1	sbcth 3417	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → [𝐴 / 𝑥](𝐵 = 𝐶 ↔ ∀𝑦(𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶)))
3		sbcbig 3447	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝐵 = 𝐶 ↔ ∀𝑦(𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶)) ↔ ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ [𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶))))
4	2, 3	mpbid 221	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ [𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶)))
5		sbcal 3452	. . . 4 ⊢ ([𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶) ↔ ∀𝑦[𝐴 / 𝑥](𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶))
6	4, 5	syl6bb 275	. . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ∀𝑦[𝐴 / 𝑥](𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶)))
7		sbcbig 3447	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐶)))
8	7	albidv 1836	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑦[𝐴 / 𝑥](𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶) ↔ ∀𝑦([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐶)))
9		sbcel2 3941	. . . . . 6 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)
10	9	a1i 11	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵))
11		sbcel2 3941	. . . . . 6 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)
12	11	a1i 11	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))
13	10, 12	bibi12d 334	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐶) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))
14	13	albidv 1836	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑦([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐶) ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))
15	6, 8, 14	3bitrd 293	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))
16		dfcleq 2604	. 2 ⊢ (⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶 ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))
17	15, 16	syl6bbr 277	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶))

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   = wceq 1475   ∈ wcel 1977  [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)