Theorem sbceqi 4374

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 4373	. . 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 2780	. 2 ⊢ (⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶 ↔ 𝐷 = 𝐸)
7	3, 6	bitri 278	1 ⊢ ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ 𝐷 = 𝐸)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2145  Vcvv 3453  [wsbc 3742  ⦋csb 3850

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-sbc 3743  df-csb 3851

This theorem is used by:  sbccom2lem  38859