Theorem excomm-P6 740

Description: Existential Quantifier Commutivity.

See excommw-P5 670 for a version that requires only FOL axioms.

Assertion

Ref	Expression
excomm-P6	⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑)

Distinct variable group:   𝑥,𝑦

Proof of Theorem excomm-P6

Step	Hyp	Ref	Expression
1		allcomm-P6 739	. . 3 ⊢ (∀𝑥∀𝑦 ¬ 𝜑 ↔ ∀𝑦∀𝑥 ¬ 𝜑)
2		lemma-L5.03a 666	. . 3 ⊢ (∀𝑥∀𝑦 ¬ 𝜑 ↔ ¬ ∃𝑥∃𝑦𝜑)
3		lemma-L5.03a 666	. . 3 ⊢ (∀𝑦∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑦∃𝑥𝜑)
4	1, 2, 3	subbid2-P4.RC 551	. 2 ⊢ (¬ ∃𝑥∃𝑦𝜑 ↔ ¬ ∃𝑦∃𝑥𝜑)
5	4	negbicancel-P4.11.RC 420	1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑)

Syntax hints:  ∀wff-forall 8  ¬ wff-neg 9   ↔ wff-bi 104  ∃wff-exists 595

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14  ax-GEN 15  ax-L4 16  ax-L11 28

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-or-D2.3 145  df-true-D2.4 155  df-rcp-AND3 161  df-exists-D5.1 596

This theorem is referenced by:  nfrex2-P6  744  nfrex2d-P6  820