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Theorem raleqbidv 2613
Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
Hypotheses
Ref Expression
raleqbidv.1 (𝜑𝐴 = 𝐵)
raleqbidv.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
raleqbidv (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem raleqbidv
StepHypRef Expression
1 raleqbidv.1 . . 3 (𝜑𝐴 = 𝐵)
21raleqdv 2607 . 2 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜓))
3 raleqbidv.2 . . 3 (𝜑 → (𝜓𝜒))
43ralbidv 2412 . 2 (𝜑 → (∀𝑥𝐵 𝜓 ↔ ∀𝑥𝐵 𝜒))
52, 4bitrd 187 1 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1314  wral 2391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396
This theorem is referenced by:  ofrfval  5956  fmpox  6064  tfrlemi1  6195  supeq123d  6844  cvg1nlemcau  10707  cvg1nlemres  10708  cau3lem  10837  fsum2dlemstep  11154  fisumcom2  11158  istopg  12072  restbasg  12243  cnfval  12269  cnpfval  12270  txbas  12333  limccl  12703  sscoll2  13020
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