Theorem bj-sbeq 37793

Description: Distribute proper substitution through an equality relation. (See sbceqg 4370). (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 2754	. . . . 5 ⊢ (𝐴 = 𝐵 ↔ ∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵))
2	1	sbbii 2113	. . . 4 ⊢ ([𝑦 / 𝑥]𝐴 = 𝐵 ↔ [𝑦 / 𝑥]∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵))
3		sbsbc 3743	. . . 4 ⊢ ([𝑦 / 𝑥]∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵) ↔ [𝑦 / 𝑥]∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵))
4		sbcal 3798	. . . 4 ⊢ ([𝑦 / 𝑥]∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵) ↔ ∀𝑧[𝑦 / 𝑥](𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵))
5	2, 3, 4	3bitri 300	. . 3 ⊢ ([𝑦 / 𝑥]𝐴 = 𝐵 ↔ ∀𝑧[𝑦 / 𝑥](𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵))
6		sbcbig 3790	. . . . 5 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑥](𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵) ↔ ([𝑦 / 𝑥]𝑧 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑧 ∈ 𝐵)))
7	6	elv 3456	. . . 4 ⊢ ([𝑦 / 𝑥](𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵) ↔ ([𝑦 / 𝑥]𝑧 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑧 ∈ 𝐵))
8	7	albii 1852	. . 3 ⊢ (∀𝑧[𝑦 / 𝑥](𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵) ↔ ∀𝑧([𝑦 / 𝑥]𝑧 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑧 ∈ 𝐵))
9		sbcel2 4376	. . . . 5 ⊢ ([𝑦 / 𝑥]𝑧 ∈ 𝐴 ↔ 𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐴)
10		sbcel2 4376	. . . . 5 ⊢ ([𝑦 / 𝑥]𝑧 ∈ 𝐵 ↔ 𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵)
11	9, 10	bibi12i 342	. . . 4 ⊢ (([𝑦 / 𝑥]𝑧 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑧 ∈ 𝐵) ↔ (𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐴 ↔ 𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵))
12	11	albii 1852	. . 3 ⊢ (∀𝑧([𝑦 / 𝑥]𝑧 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑧 ∈ 𝐵) ↔ ∀𝑧(𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐴 ↔ 𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵))
13	5, 8, 12	3bitri 300	. 2 ⊢ ([𝑦 / 𝑥]𝐴 = 𝐵 ↔ ∀𝑧(𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐴 ↔ 𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵))
14		dfcleq 2754	. 2 ⊢ (⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑦 / 𝑥⦌𝐵 ↔ ∀𝑧(𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐴 ↔ 𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵))
15	13, 14	bitr4i 281	1 ⊢ ([𝑦 / 𝑥]𝐴 = 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑦 / 𝑥⦌𝐵)

This proof depends on syntax axioms:   ↔ wb 209  ∀wal 1568   = wceq 1570  [wsb 2099   ∈ wcel 2145  Vcvv 3451  [wsbc 3739  ⦋csb 3847

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-nul 4280

This theorem is used by: (None)