Theorem rexcom4f 32725

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 2927	. . 3 ⊢ Ⅎ𝑥V
3	1, 2	rexcomf 3304	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ V 𝜑 ↔ ∃𝑦 ∈ V ∃𝑥 ∈ 𝐴 𝜑)
4		rexv 3484	. . 3 ⊢ (∃𝑦 ∈ V 𝜑 ↔ ∃𝑦𝜑)
5	4	rexbii 3112	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ V 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦𝜑)
6		rexv 3484	. 2 ⊢ (∃𝑦 ∈ V ∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝜑)
7	3, 5, 6	3bitr3i 304	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝜑)

Syntax hints:   ↔ wb 209  ∃wex 1802  Ⅎwnfc 2912  ∃wrex 3089  Vcvv 3457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-11 2194  ax-12 2215  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-nf 1807  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3080  df-rex 3090  df-v 3459

This theorem is referenced by: (None)