Theorem csbief 3888

Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)

Hypotheses

Ref	Expression
csbief.1	⊢ 𝐴 ∈ V
csbief.2	⊢ Ⅎ𝑥𝐶
csbief.3	⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)

Assertion

Ref	Expression
csbief	⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem csbief

Step	Hyp	Ref	Expression
1		csbief.1	. 2 ⊢ 𝐴 ∈ V
2		csbief.2	. . . 4 ⊢ Ⅎ𝑥𝐶
3	2	a1i 11	. . 3 ⊢ (𝐴 ∈ V → Ⅎ𝑥𝐶)
4		csbief.3	. . 3 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
5	3, 4	csbiegf 3887	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
6	1, 5	ax-mp 5	1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  Ⅎwnfc 2910  Vcvv 3455  ⦋csb 3854

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-v 3457  df-sbc 3746  df-csb 3855

This theorem is referenced by:  cbvrabcsfw  3895  csbun  4407  csbin  4408  csbdif  4487  csbif  4546  csbopab  5542  csbopabw  5543  csbima12  6083  csbcog  6300  csbiota  6531  csbriota  7384  csbov123  7456  pcmpt  16953  mpfrcl  22217  iundisj2  25689  iundisj2f  32916  iundisj2fi  33123  csbttc  37001  csbafv12g  47857  csbaovg  47900  csbafv212g  47939