Theorem rexcom4f 29906

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	⊢ Ⅎ𝑦𝐴

Assertion

Ref	Expression
rexcom4f	⊢ (∃𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem rexcom4f

Step	Hyp	Ref	Expression
1		ralcom4f.1	. . 3 ⊢ Ⅎ𝑦𝐴
2		nfcv 2934	. . 3 ⊢ Ⅎ𝑥V
3	1, 2	rexcomf 3283	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ V 𝜑 ↔ ∃𝑦 ∈ V ∃𝑥 ∈ 𝐴 𝜑)
4		rexv 3422	. . 3 ⊢ (∃𝑦 ∈ V 𝜑 ↔ ∃𝑦𝜑)
5	4	rexbii 3224	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ V 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦𝜑)
6		rexv 3422	. 2 ⊢ (∃𝑦 ∈ V ∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝜑)
7	3, 5, 6	3bitr3i 293	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝜑)

Syntax hints:   ↔ wb 198  ∃wex 1823  Ⅎwnfc 2919  ∃wrex 3091  Vcvv 3398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-v 3400

This theorem is referenced by: (None)