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Theorem rexeqbi1dv 3124
 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 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem rexeqbi1dv
StepHypRef Expression
1 rexeq 3116 . 2 (𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑))
2 raleqd.1 . . 3 (𝐴 = 𝐵 → (𝜑𝜓))
32rexbidv 3034 . 2 (𝐴 = 𝐵 → (∃𝑥𝐵 𝜑 ↔ ∃𝑥𝐵 𝜓))
41, 3bitrd 267 1 (𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  ∃wrex 2897 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902 This theorem is referenced by:  fri  5000  frsn  5112  isofrlem  6490  f1oweALT  7043  frxp  7174  1sdom  8048  oieq2  8301  zfregcl  8382  zfregclOLD  8384  ishaus  20936  isreg  20946  isnrm  20949  lebnumlem3  22570  1vwmgra  26530  3vfriswmgra  26532  isgrpo  26735  pjhth  27636  bnj1154  30321  frmin  30983  isexid2  32824  ismndo2  32843  rngomndo  32904  stoweidlem28  38921  1vwmgr  41446  3vfriswmgr  41448
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