Theorem csbiegf 3885

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 group:   𝑥,𝐴

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

Proof of Theorem csbiegf

Step	Hyp	Ref	Expression
1		csbiegf.2	. . 3 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
2	1	ax-gen 1823	. 2 ⊢ ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶)
3		csbiegf.1	. . 3 ⊢ (𝐴 ∈ 𝑉 → Ⅎ𝑥𝐶)
4		csbiebt 3881	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
5	3, 4	mpdan 699	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
6	2, 5	mpbii 236	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1566   = wceq 1568   ∈ wcel 2141  Ⅎwnfc 2908  ⦋csb 3852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-ex 1808  df-nf 1812  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-v 3455  df-sbc 3744  df-csb 3853

This theorem is referenced by:  csbief  3886  sbcco3gw  4389  sbcco3g  4394  csbco3g  4395  fmptcof  7126  fmpoco  8089  sumsnf  15793  prodsn  16015  prodsnf  16017  bpolylem  16101  pcmpt  16951  chfacfpmmulfsupp  22999  elmptrab  23963  dvfsumrlim3  26171  itgsubstlem  26186  itgsubst  26187  ifeqeqx  32854  disjunsn  32905  sbcaltop  36427  unirep  38309  cdleme31so  41099  cdleme31sn  41100  cdleme31sn1  41101  cdleme31se  41102  cdleme31se2  41103  cdleme31sc  41104  cdleme31sde  41105  cdleme31sn2  41109  cdlemeg47rv2  41230  cdlemk41  41640  monotuz  43616  oddcomabszz  43619