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Theorem csbied 3178
 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 (φA 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
StepHypRef Expression
1 nfv 1619 . 2 xφ
2 nfcvd 2490 . 2 (φxC)
3 csbied.1 . 2 (φA V)
4 csbied.2 . 2 ((φ x = A) → B = C)
51, 2, 3, 4csbiedf 3173 1 (φ[A / x]B = C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  [csb 3136 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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:  csbied2  3179  fvmptd  5702
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