Theorem csbiegf 3916

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 1796	. 2 ⊢ ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶)
3		csbiegf.1	. . 3 ⊢ (𝐴 ∈ 𝑉 → Ⅎ𝑥𝐶)
4		csbiebt 3912	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
5	3, 4	mpdan 685	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
6	2, 5	mpbii 235	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1535   = wceq 1537   ∈ wcel 2114  Ⅎwnfc 2961  ⦋csb 3883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-sbc 3773  df-csb 3884

This theorem is referenced by:  csbief  3917  sbcco3gw  4374  sbcco3g  4379  csbco3g  4380  fmptcof  6892  fmpoco  7790  sumsnf  15099  prodsn  15316  prodsnf  15318  bpolylem  15402  pcmpt  16228  chfacfpmmulfsupp  21471  elmptrab  22435  dvfsumrlim3  24630  itgsubstlem  24645  itgsubst  24646  ifeqeqx  30297  disjunsn  30344  sbcaltop  33442  unirep  35003  cdleme31so  37530  cdleme31sn  37531  cdleme31sn1  37532  cdleme31se  37533  cdleme31se2  37534  cdleme31sc  37535  cdleme31sde  37536  cdleme31sn2  37540  cdlemeg47rv2  37661  cdlemk41  38071  monotuz  39558  oddcomabszz  39561