MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rexbidv2 Structured version   Visualization version   GIF version

Theorem rexbidv2 3175
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 1921 . 2 (𝜑 → (∃𝑥(𝑥𝐴𝜓) ↔ ∃𝑥(𝑥𝐵𝜒)))
3 df-rex 3071 . 2 (∃𝑥𝐴 𝜓 ↔ ∃𝑥(𝑥𝐴𝜓))
4 df-rex 3071 . 2 (∃𝑥𝐵 𝜒 ↔ ∃𝑥(𝑥𝐵𝜒))
52, 3, 43bitr4g 314 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wex 1779  wcel 2108  wrex 3070
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 1910
This theorem depends on definitions:  df-bi 207  df-ex 1780  df-rex 3071
This theorem is referenced by:  rexbidva  3177  rexeqbidv  3347  rexssOLD  4061  iuneq12d  5021  exopxfr2  5855  isoini  7358  rexsupp  8207  omabs  8689  elfi2  9454  wemapsolem  9590  ltexpi  10942  rexuz  12940  ncoprmgcdne1b  16687  lpigen  21345  llyi  23482  nllyi  23483  elpi1  25078  ressupprn  32699  xrecex  32902  bnj18eq1  34941  ldual1dim  39167  pmapjat1  39855  mrefg2  42718  islssfg2  43083  fourierdlem71  46192  hoiqssbl  46640  reuxfr1dd  48726  lubeldm2d  48855  glbeldm2d  48856
  Copyright terms: Public domain W3C validator