Theorem sbceqi 4403

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 4402	. . 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 2742	. 2 ⊢ (⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶 ↔ 𝐷 = 𝐸)
7	3, 6	bitri 275	1 ⊢ ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ 𝐷 = 𝐸)

Syntax hints:   ↔ wb 205   = wceq 1533   ∈ wcel 2098  Vcvv 3466  [wsbc 3770  ⦋csb 3886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-sbc 3771  df-csb 3887

This theorem is referenced by:  sbccom2lem  37496