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

Theorem rexbid 3045
Description: Formula-building rule for restricted existential quantifier (deduction rule). (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 481 . 2 ((𝜑𝑥𝐴) → (𝜓𝜒))
41, 3rexbida 3041 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wnf 1705  wcel 1987  wrex 2908
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-12 2044
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-nf 1707  df-rex 2913
This theorem is referenced by:  rexbidvALT  3047  rexeqbid  3143  scott0  8700  infcvgaux1i  14521  bnj1463  30858  poimirlem25  33093  poimirlem26  33094  elrnmptf  38864  smfsupmpt  40349  smfinfmpt  40353
  Copyright terms: Public domain W3C validator