Theorem csbief 3089

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 9	. . 3 ⊢ (𝐴 ∈ V → Ⅎ𝑥𝐶)
4		csbief.3	. . 3 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
5	3, 4	csbiegf 3088	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
6	1, 5	ax-mp 5	1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶

Syntax hints:   → wi 4   = wceq 1343   ∈ wcel 2136  Ⅎwnfc 2295  Vcvv 2726  ⦋csb 3045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sbc 2952  df-csb 3046

This theorem is referenced by:  csbie  3090  csbing  3329  csbopabg  4060  pofun  4290  csbima12g  4965  csbiotag  5181  csbriotag  5810  csbov123g  5880  eqerlem  6532  zsumdc  11325