Theorem qremall-P6 722

Description: Universal Quantifier Removal (non-freeness condition).

See qremallw-P6 702 for a version that requires only FOL axioms.

Hypothesis

Ref	Expression
qremall-P6.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
qremall-P6	⊢ (∀𝑥𝜑 ↔ 𝜑)

Proof of Theorem qremall-P6

Step	Hyp	Ref	Expression
1		spec-P6 719	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		exi-P6 718	. . 3 ⊢ (𝜑 → ∃𝑥𝜑)
3		qremall-P6.1	. . . 4 ⊢ Ⅎ𝑥𝜑
4		dfnfreealt-P6 683	. . . 4 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
5	3, 4	bimpf-P4.RC 532	. . 3 ⊢ (∃𝑥𝜑 → ∀𝑥𝜑)
6	2, 5	syl-P3.24.RC 260	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
7	1, 6	rcp-NDBII0 239	1 ⊢ (∀𝑥𝜑 ↔ 𝜑)

Syntax hints:  ∀wff-forall 8   → wff-imp 10   ↔ wff-bi 104  ∃wff-exists 595  Ⅎwff-nfree 681

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-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

This theorem is referenced by:  lemma-L6.02a  726  qceximr-P6  758  qceximl-P6  760  psubnfr-P6  784