Theorem ralcom4OLD 3522

Description: Obsolete version of ralcom4 3234 as of 26-Aug-2023. Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ralcom4OLD	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦𝜑 ↔ ∀𝑦∀𝑥 ∈ 𝐴 𝜑)

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

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

Proof of Theorem ralcom4OLD

Step	Hyp	Ref	Expression
1		ralcom 3353	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ V 𝜑 ↔ ∀𝑦 ∈ V ∀𝑥 ∈ 𝐴 𝜑)
2		ralv 3516	. . 3 ⊢ (∀𝑦 ∈ V 𝜑 ↔ ∀𝑦𝜑)
3	2	ralbii 3164	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ V 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦𝜑)
4		ralv 3516	. 2 ⊢ (∀𝑦 ∈ V ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦∀𝑥 ∈ 𝐴 𝜑)
5	1, 3, 4	3bitr3i 303	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦𝜑 ↔ ∀𝑦∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   ↔ wb 208  ∀wal 1534  ∀wral 3137  Vcvv 3491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-11 2160  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-ral 3142  df-v 3493

This theorem is referenced by: (None)