Theorem qremall-P8 1101

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

Hypothesis

Ref	Expression
qremall-P8.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
qremall-P8	⊢ (∀𝑥𝜑 ↔ 𝜑)

Proof of Theorem qremall-P8

Step	Hyp	Ref	Expression
1		alle-P7.CL 942	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		qremall-P8.1	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	nfrgen-P7.CL 930	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
4	1, 3	rcp-NDBII0 239	1 ⊢ (∀𝑥𝜑 ↔ 𝜑)

Syntax hints:  ∀wff-forall 8   ↔ wff-bi 104  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-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:  qremall-P8.VR  1102  qceximr-P8  1123  qceximl-P8  1125