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Theorem rexeqbi1dv 2508
Description: Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.)
Hypothesis
Ref Expression
raleqd.1 (A = B → (φψ))
Assertion
Ref Expression
rexeqbi1dv (A = B → (x A φx B ψ))
Distinct variable groups:   x,A   x,B
Allowed substitution hints:   φ(x)   ψ(x)

Proof of Theorem rexeqbi1dv
StepHypRef Expression
1 rexeq 2500 . 2 (A = B → (x A φx B φ))
2 raleqd.1 . . 3 (A = B → (φψ))
32rexbidv 2321 . 2 (A = B → (x B φx B ψ))
41, 3bitrd 177 1 (A = B → (x A φx 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:  bj-nn0suc0  9408
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