Theorem bj-sbeq 36247

Description: Distribute proper substitution through an equality relation. (See sbceqg 4409). (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 2724	. . . . 5 ⊢ (𝐴 = 𝐵 ↔ ∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵))
2	1	sbbii 2078	. . . 4 ⊢ ([𝑦 / 𝑥]𝐴 = 𝐵 ↔ [𝑦 / 𝑥]∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵))
3		sbsbc 3781	. . . 4 ⊢ ([𝑦 / 𝑥]∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵) ↔ [𝑦 / 𝑥]∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵))
4		sbcal 3841	. . . 4 ⊢ ([𝑦 / 𝑥]∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵) ↔ ∀𝑧[𝑦 / 𝑥](𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵))
5	2, 3, 4	3bitri 297	. . 3 ⊢ ([𝑦 / 𝑥]𝐴 = 𝐵 ↔ ∀𝑧[𝑦 / 𝑥](𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵))
6		sbcbig 3831	. . . . 5 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑥](𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵) ↔ ([𝑦 / 𝑥]𝑧 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑧 ∈ 𝐵)))
7	6	elv 3479	. . . 4 ⊢ ([𝑦 / 𝑥](𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵) ↔ ([𝑦 / 𝑥]𝑧 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑧 ∈ 𝐵))
8	7	albii 1820	. . 3 ⊢ (∀𝑧[𝑦 / 𝑥](𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵) ↔ ∀𝑧([𝑦 / 𝑥]𝑧 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑧 ∈ 𝐵))
9		sbcel2 4415	. . . . 5 ⊢ ([𝑦 / 𝑥]𝑧 ∈ 𝐴 ↔ 𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐴)
10		sbcel2 4415	. . . . 5 ⊢ ([𝑦 / 𝑥]𝑧 ∈ 𝐵 ↔ 𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵)
11	9, 10	bibi12i 339	. . . 4 ⊢ (([𝑦 / 𝑥]𝑧 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑧 ∈ 𝐵) ↔ (𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐴 ↔ 𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵))
12	11	albii 1820	. . 3 ⊢ (∀𝑧([𝑦 / 𝑥]𝑧 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑧 ∈ 𝐵) ↔ ∀𝑧(𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐴 ↔ 𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵))
13	5, 8, 12	3bitri 297	. 2 ⊢ ([𝑦 / 𝑥]𝐴 = 𝐵 ↔ ∀𝑧(𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐴 ↔ 𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵))
14		dfcleq 2724	. 2 ⊢ (⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑦 / 𝑥⦌𝐵 ↔ ∀𝑧(𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐴 ↔ 𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵))
15	13, 14	bitr4i 278	1 ⊢ ([𝑦 / 𝑥]𝐴 = 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑦 / 𝑥⦌𝐵)

Syntax hints:   ↔ wb 205  ∀wal 1538   = wceq 1540  [wsb 2066   ∈ wcel 2105  Vcvv 3473  [wsbc 3777  ⦋csb 3893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-nul 4323

This theorem is referenced by: (None)