Theorem idempotexnall-P8 1100

Description: Idempotency Law for '∃𝑥' over '¬ ∀𝑥'. †

Assertion

Ref	Expression
idempotexnall-P8	⊢ (∃𝑥 ¬ ∀𝑥𝜑 ↔ ¬ ∀𝑥𝜑)

Proof of Theorem idempotexnall-P8

Step	Hyp	Ref	Expression
1		ndnfrall1-P7.7 832	. . . 4 ⊢ Ⅎ𝑥∀𝑥𝜑
2	1	ndnfrneg-P7.2.RC 875	. . 3 ⊢ Ⅎ𝑥 ¬ ∀𝑥𝜑
3	2	nfrexgen-P7.CL 932	. 2 ⊢ (∃𝑥 ¬ ∀𝑥𝜑 → ¬ ∀𝑥𝜑)
4		exi-P7.CL 952	. 2 ⊢ (¬ ∀𝑥𝜑 → ∃𝑥 ¬ ∀𝑥𝜑)
5	3, 4	rcp-NDBII0 239	1 ⊢ (∃𝑥 ¬ ∀𝑥𝜑 ↔ ¬ ∀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-L5 17  ax-L6 18  ax-L7 19  ax-L10 27  ax-L11 28  ax-L12 29

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  df-nfree-D6.1 682  df-psub-D6.2 716

This theorem is referenced by: (None)