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Theorem ceqsrexbv 2973
 Description: Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by Mario Carneiro, 14-Mar-2014.)
Hypothesis
Ref Expression
ceqsrexv.1 (x = A → (φψ))
Assertion
Ref Expression
ceqsrexbv (x B (x = A φ) ↔ (A B ψ))
Distinct variable groups:   x,A   x,B   ψ,x
Allowed substitution hint:   φ(x)

Proof of Theorem ceqsrexbv
StepHypRef Expression
1 r19.42v 2765 . 2 (x B (A B (x = A φ)) ↔ (A B x B (x = A φ)))
2 eleq1 2413 . . . . . . 7 (x = A → (x BA B))
32adantr 451 . . . . . 6 ((x = A φ) → (x BA B))
43pm5.32ri 619 . . . . 5 ((x B (x = A φ)) ↔ (A B (x = A φ)))
54bicomi 193 . . . 4 ((A B (x = A φ)) ↔ (x B (x = A φ)))
65baib 871 . . 3 (x B → ((A B (x = A φ)) ↔ (x = A φ)))
76rexbiia 2647 . 2 (x B (A B (x = A φ)) ↔ x B (x = A φ))
8 ceqsrexv.1 . . . 4 (x = A → (φψ))
98ceqsrexv 2972 . . 3 (A B → (x B (x = A φ) ↔ ψ))
109pm5.32i 618 . 2 ((A B x B (x = A φ)) ↔ (A B ψ))
111, 7, 103bitr3i 266 1 (x B (x = A φ) ↔ (A B ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2615 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-rex 2620  df-v 2861 This theorem is referenced by: (None)
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