Theorem csbiegf 3171

Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 11-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)

Hypotheses

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

Assertion

Ref	Expression
csbiegf	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑉

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

Proof of Theorem csbiegf

Step	Hyp	Ref	Expression
1		csbiegf.2	. . 3 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
2	1	ax-gen 1497	. 2 ⊢ ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶)
3		csbiegf.1	. . 3 ⊢ (𝐴 ∈ 𝑉 → Ⅎ𝑥𝐶)
4		csbiebt 3167	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
5	3, 4	mpdan 421	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
6	2, 5	mpbii 148	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1395   = wceq 1397   ∈ wcel 2202  Ⅎwnfc 2361  ⦋csb 3127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-sbc 3032  df-csb 3128

This theorem is referenced by:  csbief  3172  sbcco3g  3185  csbco3g  3186  fmptcof  5814  fmpoco  6381  iseqf1olemjpcl  10771  iseqf1olemqpcl  10772  iseqf1olemfvp  10773  seq3f1olemqsum  10776  sumsnf  11972  prodsnf  12155  pcmpt  12918