Theorem sbceqi 4436

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 4435	. . 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 2758	. 2 ⊢ (⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶 ↔ 𝐷 = 𝐸)
7	3, 6	bitri 275	1 ⊢ ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ 𝐷 = 𝐸)

Syntax hints:   ↔ wb 206   = wceq 1537   ∈ wcel 2108  Vcvv 3488  [wsbc 3804  ⦋csb 3921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-sbc 3805  df-csb 3922

This theorem is referenced by:  sbccom2lem  38084