Theorem csbief 3177

Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)

Hypotheses

Ref	Expression
csbief.1	⊢ A ∈ V
csbief.2	⊢ ℲxC
csbief.3	⊢ (x = A → B = C)

Assertion

Ref	Expression
csbief	⊢ [A / x]B = C

Distinct variable group:   x,A

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

Proof of Theorem csbief

Step	Hyp	Ref	Expression
1		csbief.1	. 2 ⊢ A ∈ V
2		csbief.2	. . . 4 ⊢ ℲxC
3	2	a1i 10	. . 3 ⊢ (A ∈ V → ℲxC)
4		csbief.3	. . 3 ⊢ (x = A → B = C)
5	3, 4	csbiegf 3176	. 2 ⊢ (A ∈ V → [A / x]B = C)
6	1, 5	ax-mp 5	1 ⊢ [A / x]B = C

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476  Vcvv 2859  [csb 3136

This theorem was proved from 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 theorem 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 2478  df-v 2861  df-sbc 3047  df-csb 3137

This theorem is referenced by:  csbing  3462  csbifg  3690  csbiotag  4371  csbopabg  4637  csbima12g  4955  csbovg  5552  eqerlem  5960