Theorem lemma-L5.03a 666

Description: A Lemma for Universal / Existential Conversion of Nested Quantifiers.

Assertion

Ref	Expression
lemma-L5.03a	⊢ (∀𝑥∀𝑦 ¬ 𝜑 ↔ ¬ ∃𝑥∃𝑦𝜑)

Distinct variable group:   𝑥,𝑦

Proof of Theorem lemma-L5.03a

Step	Hyp	Ref	Expression
1		allnegex-P5 597	. . 3 ⊢ (∀𝑦 ¬ 𝜑 ↔ ¬ ∃𝑦𝜑)
2	1	suballinf-P5 594	. 2 ⊢ (∀𝑥∀𝑦 ¬ 𝜑 ↔ ∀𝑥 ¬ ∃𝑦𝜑)
3		allnegex-P5 597	. 2 ⊢ (∀𝑥 ¬ ∃𝑦𝜑 ↔ ¬ ∃𝑥∃𝑦𝜑)
4	2, 3	bitrns-P3.33c.RC 303	1 ⊢ (∀𝑥∀𝑦 ¬ 𝜑 ↔ ¬ ∃𝑥∃𝑦𝜑)

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

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:  lemma-L5.05a  668  excomm-P6  740