Theorem allcomm-P6 739

Description: Universal Quantifier Commutivity.

See allcommw-P5 669 for a version that requires only FOL axioms.

Assertion

Ref	Expression
allcomm-P6	⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑦∀𝑥𝜑)

Distinct variable group:   𝑥,𝑦

Proof of Theorem allcomm-P6

Step	Hyp	Ref	Expression
1		ax-L11 28	. 2 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
2		ax-L11 28	. 2 ⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑)
3	1, 2	rcp-NDBII0 239	1 ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑦∀𝑥𝜑)

Syntax hints:  ∀wff-forall 8   ↔ wff-bi 104

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14  ax-L11 28

This theorem depends on definitions:  df-bi-D2.1 107  df-true-D2.4 155

This theorem is referenced by:  excomm-P6  740  nfrall2-P6  743  nfrall2d-P6  819