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Theorem rexcom4f 28507
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 2750 . . 3 𝑥V
31, 2rexcomf 3077 . 2 (∃𝑥𝐴𝑦 ∈ V 𝜑 ↔ ∃𝑦 ∈ V ∃𝑥𝐴 𝜑)
4 rexv 3192 . . 3 (∃𝑦 ∈ V 𝜑 ↔ ∃𝑦𝜑)
54rexbii 3022 . 2 (∃𝑥𝐴𝑦 ∈ V 𝜑 ↔ ∃𝑥𝐴𝑦𝜑)
6 rexv 3192 . 2 (∃𝑦 ∈ V ∃𝑥𝐴 𝜑 ↔ ∃𝑦𝑥𝐴 𝜑)
73, 5, 63bitr3i 288 1 (∃𝑥𝐴𝑦𝜑 ↔ ∃𝑦𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 194  wex 1694  wnfc 2737  wrex 2896  Vcvv 3172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-v 3174
This theorem is referenced by: (None)
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