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Theorem raleqf 2479
Description: Equality theorem for restricted universal quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
Hypotheses
Ref Expression
raleq1f.1 xA
raleq1f.2 xB
Assertion
Ref Expression
raleqf (A = B → (x A φx B φ))

Proof of Theorem raleqf
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))
54imbi1d 220 . . 3 (A = B → ((x Aφ) ↔ (x Bφ)))
63, 5albid 1488 . 2 (A = B → (x(x Aφ) ↔ x(x Bφ)))
7 df-ral 2289 . 2 (x A φx(x Aφ))
8 df-ral 2289 . 2 (x B φx(x Bφ))
96, 7, 83bitr4g 212 1 (A = B → (x A φx B φ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1226   = wceq 1228   wcel 1374  wnfc 2147  wral 2284
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-ral 2289
This theorem is referenced by:  raleq  2483  repizf2  3889
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