Theorem csbief 3700

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 3699	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
6	1, 5	ax-mp 5	1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶

Syntax hints:   → wi 4   = wceq 1632   ∈ wcel 2140  Ⅎwnfc 2890  Vcvv 3341  ⦋csb 3675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-v 3343  df-sbc 3578  df-csb 3676

This theorem is referenced by:  csbie  3701  csbun  4153  csbin  4154  csbif  4283  csbopab  5159  csbopabgALT  5160  csbima12  5642  csbiota  6043  csbriota  6788  csbov123  6852  pcmpt  15819  mpfrcl  19741  iundisj2  23538  iundisj2f  29732  iundisj2fi  29887  csbdif  33501  sbccom2f  34263  csbcog  38462  csbingOLD  39573  csbafv12g  41742  csbaovg  41785