Theorem dfnfreealt-P6 683

Description: Alternate Definition of Effective Non-Freeness.

Assertion

Ref	Expression
dfnfreealt-P6	⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))

Proof of Theorem dfnfreealt-P6

Step	Hyp	Ref	Expression
1		df-nfree-D6.1 682	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
2		allnegex-P5 597	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
3	2	suborr-P3.43b.RC 349	. 2 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
4		orcomm-P3.37 319	. 2 ⊢ ((∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑) ↔ (¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑))
5		imasor-P4.32a 487	. . 3 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑))
6	5	bisym-P3.33b.RC 299	. 2 ⊢ ((¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑) ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
7	1, 3, 4, 6	tbitrns-P4.17.RC 431	1 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))

Syntax hints:  ∀wff-forall 8  ¬ wff-neg 9   → wff-imp 10   ↔ wff-bi 104   ∨ wff-or 144  ∃wff-exists 595  Ⅎwff-nfree 681

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  df-nfree-D6.1 682

This theorem is referenced by:  nfrgenw-P6  684  gennfrw-P6  685  nfrim-P6  689  qimeqalla-P6-L1  698  qimeqallb-P6-L1  700  qremall-P6  722  qremex-P6  723  nfrgen-P6  733  gennfr-P6  734  subnfr-P6  755  nfrgencl-L6  811  gennfrcl-L6  812  nfrimd-P6  815  nfrnfr-P6  821  qremexd-P6  823