Theorem sbceqi 4318

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 4317	. . 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 2813	. 2 ⊢ (⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶 ↔ 𝐷 = 𝐸)
7	3, 6	bitri 278	1 ⊢ ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ 𝐷 = 𝐸)

Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2111  Vcvv 3441  [wsbc 3720  ⦋csb 3828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-sbc 3721  df-csb 3829

This theorem is referenced by:  sbccom2lem  35562