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Theorem rexeqbidv 2535
 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
rexeqbidv (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem rexeqbidv
StepHypRef Expression
1 raleqbidv.1 . . 3 (𝜑𝐴 = 𝐵)
21rexeqdv 2529 . 2 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜓))
3 raleqbidv.2 . . 3 (𝜑 → (𝜓𝜒))
43rexbidv 2344 . 2 (𝜑 → (∃𝑥𝐵 𝜓 ↔ ∃𝑥𝐵 𝜒))
52, 4bitrd 181 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 102   = wceq 1259  ∃wrex 2324 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329 This theorem is referenced by:  supeq123d  6396
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