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Theorem rexbidv2 3172
Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 22-May-1999.)
Hypothesis
Ref Expression
rexbidv2.1 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))
Assertion
Ref Expression
rexbidv2 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem rexbidv2
StepHypRef Expression
1 rexbidv2.1 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))
21exbidv 1922 . 2 (𝜑 → (∃𝑥(𝑥𝐴𝜓) ↔ ∃𝑥(𝑥𝐵𝜒)))
3 df-rex 3069 . 2 (∃𝑥𝐴 𝜓 ↔ ∃𝑥(𝑥𝐴𝜓))
4 df-rex 3069 . 2 (∃𝑥𝐵 𝜒 ↔ ∃𝑥(𝑥𝐵𝜒))
52, 3, 43bitr4g 313 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wex 1779  wcel 2104  wrex 3068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911
This theorem depends on definitions:  df-bi 206  df-ex 1780  df-rex 3069
This theorem is referenced by:  rexbidva  3174  rexeqbidv  3341  rexss  4054  exopxfr2  5843  isoini  7337  rexsupp  8169  omabs  8652  elfi2  9411  wemapsolem  9547  ltexpi  10899  rexuz  12886  ncoprmgcdne1b  16591  lpigen  21094  llyi  23198  nllyi  23199  elpi1  24792  ressupprn  32179  xrecex  32353  bnj18eq1  34236  ldual1dim  38339  pmapjat1  39027  mrefg2  41747  islssfg2  42115  fourierdlem71  45191  hoiqssbl  45639  lubeldm2d  47678  glbeldm2d  47679
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