Theorem sbceqi 4349

Description: Distribution of class substitution over equality, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)

Hypotheses

Ref	Expression
sbceqi.1	⊢ 𝐴 ∈ V
sbceqi.2	⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐷
sbceqi.3	⊢ ⦋𝐴 / 𝑥⦌𝐶 = 𝐸

Assertion

Ref	Expression
sbceqi	⊢ ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ 𝐷 = 𝐸)

Proof of Theorem sbceqi

Step	Hyp	Ref	Expression
1		sbceqi.1	. . 3 ⊢ 𝐴 ∈ V
2		sbceqg 4348	. . 3 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶))
3	1, 2	ax-mp 5	. 2 ⊢ ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)
4		sbceqi.2	. . 3 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐷
5		sbceqi.3	. . 3 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = 𝐸
6	4, 5	eqeq12i 2757	. 2 ⊢ (⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶 ↔ 𝐷 = 𝐸)
7	3, 6	bitri 274	1 ⊢ ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ 𝐷 = 𝐸)

Syntax hints:   ↔ wb 205   = wceq 1541   ∈ wcel 2109  Vcvv 3430  [wsbc 3719  ⦋csb 3836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1544  df-ex 1786  df-nf 1790  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-sbc 3720  df-csb 3837

This theorem is referenced by:  sbccom2lem  36261