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

Theorem rexeqf 3274
Description: Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 9-Oct-2003.) (Revised by Andrew Salmon, 11-Jul-2011.)
Hypotheses
Ref Expression
raleq1f.1 𝑥𝐴
raleq1f.2 𝑥𝐵
Assertion
Ref Expression
rexeqf (𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑))

Proof of Theorem rexeqf
StepHypRef Expression
1 raleq1f.1 . . . 4 𝑥𝐴
2 raleq1f.2 . . . 4 𝑥𝐵
31, 2nfeq 2914 . . 3 𝑥 𝐴 = 𝐵
4 eleq2 2828 . . . 4 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
54anbi1d 743 . . 3 (𝐴 = 𝐵 → ((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜑)))
63, 5exbid 2238 . 2 (𝐴 = 𝐵 → (∃𝑥(𝑥𝐴𝜑) ↔ ∃𝑥(𝑥𝐵𝜑)))
7 df-rex 3056 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
8 df-rex 3056 . 2 (∃𝑥𝐵 𝜑 ↔ ∃𝑥(𝑥𝐵𝜑))
96, 7, 83bitr4g 303 1 (𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wex 1853  wcel 2139  wnfc 2889  wrex 3051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rex 3056
This theorem is referenced by:  rexeq  3278  rexeqbid  3290  zfrep6  7300  iuneq12daf  29701  indexa  33859
  Copyright terms: Public domain W3C validator