Theorem csbgfi 33935

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 3534	. . . 4 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
2	1	abeq2i 2735	. . 3 ⊢ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐵)
3		csbgfi.1	. . . 4 ⊢ 𝐴 ∈ V
4		csbgfi.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
5	4	nfcri 2758	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
6	3, 5	sbcgfi 33933	. . 3 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵)
7	2, 6	bitri 264	. 2 ⊢ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑦 ∈ 𝐵)
8	7	eqriv 2619	1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐵

Syntax hints:   = wceq 1483   ∈ wcel 1990  Ⅎwnfc 2751  Vcvv 3200  [wsbc 3435  ⦋csb 3533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436  df-csb 3534

This theorem is referenced by: (None)