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

Theorem rexbid 3236
Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 27-Jun-1998.)
Hypotheses
Ref Expression
rexbid.1 𝑥𝜑
rexbid.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
rexbid (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))

Proof of Theorem rexbid
StepHypRef Expression
1 rexbid.1 . 2 𝑥𝜑
2 rexbid.2 . . 3 (𝜑 → (𝜓𝜒))
32adantr 474 . 2 ((𝜑𝑥𝐴) → (𝜓𝜒))
41, 3rexbida 3232 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wnf 1827  wcel 2107  wrex 3091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-12 2163
This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824  df-nf 1828  df-rex 3096
This theorem is referenced by:  rexbidvALT  3238  rexeqbid  3347  scott0  9046  infcvgaux1i  14993  bnj1463  31722  poimirlem25  34060  poimirlem26  34061  elrnmptf  40290  smfsupmpt  41948  smfinfmpt  41952
  Copyright terms: Public domain W3C validator