Theorem csbiegf 3928

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 1798	. 2 ⊢ ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶)
3		csbiegf.1	. . 3 ⊢ (𝐴 ∈ 𝑉 → Ⅎ𝑥𝐶)
4		csbiebt 3924	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
5	3, 4	mpdan 686	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
6	2, 5	mpbii 232	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1540   = wceq 1542   ∈ wcel 2107  Ⅎwnfc 2884  ⦋csb 3894

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-v 3477  df-sbc 3779  df-csb 3895

This theorem is referenced by:  csbief  3929  sbcco3gw  4423  sbcco3g  4428  csbco3g  4429  fmptcof  7128  fmpoco  8081  sumsnf  15689  prodsn  15906  prodsnf  15908  bpolylem  15992  pcmpt  16825  chfacfpmmulfsupp  22365  elmptrab  23331  dvfsumrlim3  25550  itgsubstlem  25565  itgsubst  25566  ifeqeqx  31774  disjunsn  31825  sbcaltop  34953  unirep  36582  cdleme31so  39250  cdleme31sn  39251  cdleme31sn1  39252  cdleme31se  39253  cdleme31se2  39254  cdleme31sc  39255  cdleme31sde  39256  cdleme31sn2  39260  cdlemeg47rv2  39381  cdlemk41  39791  monotuz  41680  oddcomabszz  41683