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Theorem rexeqf 2480
Description: Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 9-Oct-2003.) (Revised by Andrew Salmon, 11-Jul-2011.)
Hypotheses
Ref Expression
raleq1f.1 xA
raleq1f.2 xB
Assertion
Ref Expression
rexeqf (A = B → (x A φx B φ))

Proof of Theorem rexeqf
StepHypRef Expression
1 raleq1f.1 . . . 4 xA
2 raleq1f.2 . . . 4 xB
31, 2nfeq 2167 . . 3 x A = B
4 eleq2 2083 . . . 4 (A = B → (x Ax B))
54anbi1d 441 . . 3 (A = B → ((x A φ) ↔ (x B φ)))
63, 5exbid 1489 . 2 (A = B → (x(x A φ) ↔ x(x B φ)))
7 df-rex 2290 . 2 (x A φx(x A φ))
8 df-rex 2290 . 2 (x B φx(x B φ))
96, 7, 83bitr4g 212 1 (A = B → (x A φx B φ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228  wex 1362   wcel 1374  wnfc 2147  wrex 2285
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290
This theorem is referenced by:  rexeq  2484
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