Theorem csbiegf 3177

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 1546	. 2 ⊢ ∀x(x = A → B = C)
3		csbiegf.1	. . 3 ⊢ (A ∈ V → ℲxC)
4		csbiebt 3173	. . 3 ⊢ ((A ∈ V ∧ ℲxC) → (∀x(x = A → B = C) ↔ [A / x]B = C))
5	3, 4	mpdan 649	. 2 ⊢ (A ∈ V → (∀x(x = A → B = C) ↔ [A / x]B = C))
6	2, 5	mpbii 202	1 ⊢ (A ∈ V → [A / x]B = C)

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2477  [csb 3137

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048  df-csb 3138

This theorem is used by:  csbief  3178  sbcco3g  3192  csbco3g  3194