Theorem qcallimr-P6 757

Description: Quantifier Collection Law: Universal Quantifier Right on Implication (non-freeness condition).

Hypothesis

Ref	Expression
qcallimr-P6.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
qcallimr-P6	⊢ ((𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓))

Proof of Theorem qcallimr-P6

Step	Hyp	Ref	Expression
1		qcallimr-P6.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2	1	qimeqalla-P6 699	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))
3	2	bisym-P3.33b.RC 299	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓))
4	1	qremex-P6 723	. . 3 ⊢ (∃𝑥𝜑 ↔ 𝜑)
5	4	subiml-P3.40a.RC 326	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ (𝜑 → ∀𝑥𝜓))
6	3, 5	subbil2-P4.RC 547	1 ⊢ ((𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓))

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:  qcexandr-P6  761