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Theorem ceqsalv 2552
 Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)
Hypotheses
Ref Expression
ceqsalv.1 A V
ceqsalv.2 (x = A → (φψ))
Assertion
Ref Expression
ceqsalv (x(x = Aφ) ↔ ψ)
Distinct variable groups:   x,A   ψ,x
Allowed substitution hint:   φ(x)

Proof of Theorem ceqsalv
StepHypRef Expression
1 nfv 1394 . 2 xψ
2 ceqsalv.1 . 2 A V
3 ceqsalv.2 . 2 (x = A → (φψ))
41, 2, 3ceqsal 2551 1 (x(x = Aφ) ↔ ψ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1221   = wceq 1223   ∈ wcel 1366  Vcvv 2526 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1309  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-ext 1995 This theorem depends on definitions:  df-bi 110  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-v 2528 This theorem is referenced by:  gencbval  2570  clel2  2645  clel4  2648  reu8  2705  raliunxp  4392  fv3  5110
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