Theorem lemma-L5.01a 600

Description: Transpositional Quantifier Equivalence Lemma.

Assertion

Ref	Expression
lemma-L5.01a	⊢ ((∃𝑥𝜑 → 𝜑) ↔ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))

Proof of Theorem lemma-L5.01a

Step	Hyp	Ref	Expression
1		df-exists-D5.1 596	. . 3 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
2	1	subiml-P3.40a.RC 326	. 2 ⊢ ((∃𝑥𝜑 → 𝜑) ↔ (¬ ∀𝑥 ¬ 𝜑 → 𝜑))
3		trnspeq-P4b 536	. 2 ⊢ ((¬ ∀𝑥 ¬ 𝜑 → 𝜑) ↔ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
4	2, 3	bitrns-P3.33c.RC 303	1 ⊢ ((∃𝑥𝜑 → 𝜑) ↔ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))

Syntax hints:  ∀wff-forall 8  ¬ wff-neg 9   → wff-imp 10   ↔ wff-bi 104  ∃wff-exists 595

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

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:  axL5ex-P5  613  exgenallw-P6  680  nfrex2w-P6  695  nfrexgenw-P6  696  exgennfrw-P6  697  exgenall-P6  732  nfrexgen-P6  735  exgennfr-P6  736  exgennfrcl-L6  814