Theorem ralcom4f 30797

Description: Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-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
ralcom4f	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦𝜑 ↔ ∀𝑦∀𝑥 ∈ 𝐴 𝜑)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem ralcom4f

Step	Hyp	Ref	Expression
1		ralcom4f.1	. . 3 ⊢ Ⅎ𝑦𝐴
2		nfcv 2908	. . 3 ⊢ Ⅎ𝑥V
3	1, 2	ralcomf 3284	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ V 𝜑 ↔ ∀𝑦 ∈ V ∀𝑥 ∈ 𝐴 𝜑)
4		ralv 3454	. . 3 ⊢ (∀𝑦 ∈ V 𝜑 ↔ ∀𝑦𝜑)
5	4	ralbii 3092	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ V 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦𝜑)
6		ralv 3454	. 2 ⊢ (∀𝑦 ∈ V ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦∀𝑥 ∈ 𝐴 𝜑)
7	3, 5, 6	3bitr3i 300	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦𝜑 ↔ ∀𝑦∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   ↔ wb 205  ∀wal 1539  Ⅎwnfc 2888  ∀wral 3065  Vcvv 3430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-11 2157  ax-12 2174  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1544  df-ex 1786  df-nf 1790  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ral 3070  df-v 3432

This theorem is referenced by: (None)