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Theorem ceqsexg 2970
 Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 11-Oct-2004.)
Hypotheses
Ref Expression
ceqsexg.1 xψ
ceqsexg.2 (x = A → (φψ))
Assertion
Ref Expression
ceqsexg (A V → (x(x = A φ) ↔ ψ))
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   ψ(x)   V(x)

Proof of Theorem ceqsexg
StepHypRef Expression
1 nfcv 2489 . 2 xA
2 nfe1 1732 . . 3 xx(x = A φ)
3 ceqsexg.1 . . 3 xψ
42, 3nfbi 1834 . 2 x(x(x = A φ) ↔ ψ)
5 ceqex 2969 . . 3 (x = A → (φx(x = A φ)))
6 ceqsexg.2 . . 3 (x = A → (φψ))
75, 6bibi12d 312 . 2 (x = A → ((φφ) ↔ (x(x = A φ) ↔ ψ)))
8 biid 227 . 2 (φφ)
91, 4, 7, 8vtoclgf 2913 1 (A V → (x(x = A φ) ↔ ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710 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-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 This theorem is referenced by:  ceqsexgv  2971
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