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Theorem ceqsrexbv 2669
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 2461 . 2 (x B (A B (x = A φ)) ↔ (A B x B (x = A φ)))
2 eleq1 2097 . . . . . . 7 (x = A → (x BA B))
32adantr 261 . . . . . 6 ((x = A φ) → (x BA B))
43pm5.32ri 428 . . . . 5 ((x B (x = A φ)) ↔ (A B (x = A φ)))
54bicomi 123 . . . 4 ((A B (x = A φ)) ↔ (x B (x = A φ)))
65baib 827 . . 3 (x B → ((A B (x = A φ)) ↔ (x = A φ)))
76rexbiia 2333 . 2 (x B (A B (x = A φ)) ↔ x B (x = A φ))
8 ceqsrexv.1 . . . 4 (x = A → (φψ))
98ceqsrexv 2668 . . 3 (A B → (x B (x = A φ) ↔ ψ))
109pm5.32i 427 . 2 ((A B x B (x = A φ)) ↔ (A B ψ))
111, 7, 103bitr3i 199 1 (x B (x = A φ) ↔ (A B ψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  wrex 2301
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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553
This theorem is referenced by:  frecsuclem3  5929
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