Theorem bj-sbceqgALT 35777

Description: Distribute proper substitution through an equality relation. Alternate proof of sbceqg 4409. (Contributed by BJ, 6-Oct-2018.) Proof modification is discouraged to avoid using sbceqg 4409, 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	⊢ (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 2725	. . . . . 6 ⊢ (𝐵 = 𝐶 ↔ ∀𝑦(𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶))
2	1	sbcth 3792	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → [𝐴 / 𝑥](𝐵 = 𝐶 ↔ ∀𝑦(𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶)))
3		sbcbig 3831	. . . . 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 231	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ [𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶)))
5		sbcal 3841	. . . 4 ⊢ ([𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶) ↔ ∀𝑦[𝐴 / 𝑥](𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶))
6	4, 5	bitrdi 286	. . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ∀𝑦[𝐴 / 𝑥](𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶)))
7		sbcbig 3831	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐶)))
8	7	albidv 1923	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑦[𝐴 / 𝑥](𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶) ↔ ∀𝑦([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐶)))
9		sbcel2 4415	. . . . . 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 4415	. . . . . 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 345	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐶) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))
14	13	albidv 1923	. . 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 304	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))
16		dfcleq 2725	. 2 ⊢ (⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶 ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))
17	15, 16	bitr4di 288	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1539   = wceq 1541   ∈ wcel 2106  [wsbc 3777  ⦋csb 3893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-nul 4323

This theorem is referenced by: (None)