Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rexcom4f Structured version   Visualization version   GIF version

Theorem rexcom4f 29445
Description: Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by Thierry Arnoux, 8-Mar-2017.)
Hypothesis
Ref Expression
ralcom4f.1 𝑦𝐴
Assertion
Ref Expression
rexcom4f (∃𝑥𝐴𝑦𝜑 ↔ ∃𝑦𝑥𝐴 𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem rexcom4f
StepHypRef Expression
1 ralcom4f.1 . . 3 𝑦𝐴
2 nfcv 2793 . . 3 𝑥V
31, 2rexcomf 3126 . 2 (∃𝑥𝐴𝑦 ∈ V 𝜑 ↔ ∃𝑦 ∈ V ∃𝑥𝐴 𝜑)
4 rexv 3251 . . 3 (∃𝑦 ∈ V 𝜑 ↔ ∃𝑦𝜑)
54rexbii 3070 . 2 (∃𝑥𝐴𝑦 ∈ V 𝜑 ↔ ∃𝑥𝐴𝑦𝜑)
6 rexv 3251 . 2 (∃𝑦 ∈ V ∃𝑥𝐴 𝜑 ↔ ∃𝑦𝑥𝐴 𝜑)
73, 5, 63bitr3i 290 1 (∃𝑥𝐴𝑦𝜑 ↔ ∃𝑦𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wex 1744  wnfc 2780  wrex 2942  Vcvv 3231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator