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Theorem rexeqdv 2506
 Description: Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.)
Hypothesis
Ref Expression
raleq1d.1 (φA = B)
Assertion
Ref Expression
rexeqdv (φ → (x A ψx B ψ))
Distinct variable groups:   x,A   x,B
Allowed substitution hints:   φ(x)   ψ(x)

Proof of Theorem rexeqdv
StepHypRef Expression
1 raleq1d.1 . 2 (φA = B)
2 rexeq 2500 . 2 (A = B → (x A ψx B ψ))
31, 2syl 14 1 (φ → (x A ψx B ψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242  ∃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-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306 This theorem is referenced by:  rexeqbidv  2512  rexeqbidva  2514  fnunirn  5349  cbvexfo  5369  genipv  6492
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