Theorem csbied 3179

Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Mario Carneiro, 13-Oct-2016.)

Hypotheses

Ref	Expression
csbied.1	⊢ (φ → A ∈ V)
csbied.2	⊢ ((φ ∧ x = A) → B = C)

Assertion

Ref	Expression
csbied	⊢ (φ → [A / x]B = C)

Distinct variable groups:   x,A   x,C   φ,x

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

Proof of Theorem csbied

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 ⊢ Ⅎxφ
2		nfcvd 2491	. 2 ⊢ (φ → ℲxC)
3		csbied.1	. 2 ⊢ (φ → A ∈ V)
4		csbied.2	. 2 ⊢ ((φ ∧ x = A) → B = C)
5	1, 2, 3, 4	csbiedf 3174	1 ⊢ (φ → [A / x]B = C)

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  [csb 3137

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 2479  df-v 2862  df-sbc 3048  df-csb 3138

This theorem is referenced by:  csbied2  3180  fvmptd  5703