Theorem csbgfi 3870

Description: Substitution for a variable not free in a class does not affect it, in inference form. (Contributed by Giovanni Mascellani, 4-Jun-2019.)

Hypotheses

Ref	Expression
csbgfi.1	⊢ 𝐴 ∈ V
csbgfi.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
csbgfi	⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐵

Proof of Theorem csbgfi

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-csb 3851	. . . 4 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
2	1	eqabri 2874	. . 3 ⊢ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐵)
3		csbgfi.1	. . . 4 ⊢ 𝐴 ∈ V
4		csbgfi.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
5	4	nfcri 2886	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
6	3, 5	sbcgfi 3815	. . 3 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵)
7	2, 6	bitri 275	. 2 ⊢ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑦 ∈ 𝐵)
8	7	eqriv 2728	1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐵

Syntax hints:   = wceq 1541   ∈ wcel 2111  Ⅎwnfc 2879  Vcvv 3436  [wsbc 3741  ⦋csb 3850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-sbc 3742  df-csb 3851

This theorem is referenced by:  fmptdF  32633  sbccom2f  38165  evl1gprodd  42149