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Theorem rexeqbi1dv 3140
 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 3132 . 2 (𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑))
2 raleqd.1 . . 3 (𝐴 = 𝐵 → (𝜑𝜓))
32rexbidv 3047 . 2 (𝐴 = 𝐵 → (∃𝑥𝐵 𝜑 ↔ ∃𝑥𝐵 𝜓))
41, 3bitrd 268 1 (𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   = wceq 1480  ∃wrex 2909 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2914 This theorem is referenced by:  fri  5046  frsn  5160  isofrlem  6555  f1oweALT  7112  frxp  7247  1sdom  8123  oieq2  8378  zfregcl  8459  zfregclOLD  8461  ishaus  21066  isreg  21076  isnrm  21079  lebnumlem3  22702  1vwmgr  27038  3vfriswmgr  27040  isgrpo  27239  pjhth  28140  bnj1154  30828  frmin  31493  isexid2  33325  ismndo2  33344  rngomndo  33405  stoweidlem28  39582
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