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Theorem ceqsrexv 2972
 Description: Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 30-Apr-2004.)
Hypothesis
Ref Expression
ceqsrexv.1 (x = A → (φψ))
Assertion
Ref Expression
ceqsrexv (A B → (x B (x = A φ) ↔ ψ))
Distinct variable groups:   x,A   x,B   ψ,x
Allowed substitution hint:   φ(x)

Proof of Theorem ceqsrexv
StepHypRef Expression
1 df-rex 2620 . . 3 (x B (x = A φ) ↔ x(x B (x = A φ)))
2 an12 772 . . . 4 ((x = A (x B φ)) ↔ (x B (x = A φ)))
32exbii 1582 . . 3 (x(x = A (x B φ)) ↔ x(x B (x = A φ)))
41, 3bitr4i 243 . 2 (x B (x = A φ) ↔ x(x = A (x B φ)))
5 eleq1 2413 . . . . 5 (x = A → (x BA B))
6 ceqsrexv.1 . . . . 5 (x = A → (φψ))
75, 6anbi12d 691 . . . 4 (x = A → ((x B φ) ↔ (A B ψ)))
87ceqsexgv 2971 . . 3 (A B → (x(x = A (x B φ)) ↔ (A B ψ)))
98bianabs 850 . 2 (A B → (x(x = A (x B φ)) ↔ ψ))
104, 9syl5bb 248 1 (A B → (x B (x = A φ) ↔ ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615 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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-v 2861 This theorem is referenced by:  ceqsrexbv  2973  ceqsrex2v  2974  fnasrn  5417  f1oiso  5499
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