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Theorem rexcom4f 28701
 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 2751 . . 3 𝑥V
31, 2rexcomf 3078 . 2 (∃𝑥𝐴𝑦 ∈ V 𝜑 ↔ ∃𝑦 ∈ V ∃𝑥𝐴 𝜑)
4 rexv 3193 . . 3 (∃𝑦 ∈ V 𝜑 ↔ ∃𝑦𝜑)
54rexbii 3023 . 2 (∃𝑥𝐴𝑦 ∈ V 𝜑 ↔ ∃𝑥𝐴𝑦𝜑)
6 rexv 3193 . 2 (∃𝑦 ∈ V ∃𝑥𝐴 𝜑 ↔ ∃𝑦𝑥𝐴 𝜑)
73, 5, 63bitr3i 289 1 (∃𝑥𝐴𝑦𝜑 ↔ ∃𝑦𝑥𝐴 𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195  ∃wex 1695  Ⅎwnfc 2738  ∃wrex 2897  Vcvv 3173 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175 This theorem is referenced by: (None)
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