Theorem rexcom4f 32852

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 2928	. . 3 ⊢ Ⅎ𝑥V
3	1, 2	rexcomf 3307	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ V 𝜑 ↔ ∃𝑦 ∈ V ∃𝑥 ∈ 𝐴 𝜑)
4		rexv 3485	. . 3 ⊢ (∃𝑦 ∈ V 𝜑 ↔ ∃𝑦𝜑)
5	4	rexbii 3115	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ V 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦𝜑)
6		rexv 3485	. 2 ⊢ (∃𝑦 ∈ V ∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝜑)
7	3, 5, 6	3bitr3i 304	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝜑)

This proof depends on syntax axioms:   ↔ wb 209  ∃wex 1812  Ⅎwnfc 2913  ∃wrex 3092  Vcvv 3458

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-11 2195  ax-12 2216  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ral 3083  df-rex 3093  df-v 3460

This theorem is used by: (None)