Theorem qcalliml-P6 759

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

Hypothesis

Ref	Expression
qcalliml-P6.1	⊢ Ⅎ𝑥𝜓

Assertion

Ref	Expression
qcalliml-P6	⊢ ((∀𝑥𝜑 → 𝜓) ↔ ∃𝑥(𝜑 → 𝜓))

Proof of Theorem qcalliml-P6

Step	Hyp	Ref	Expression
1		qimeqex-P5 612	. . 3 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2	1	bisym-P3.33b.RC 299	. 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ ∃𝑥(𝜑 → 𝜓))
3		qcalliml-P6.1	. . . 4 ⊢ Ⅎ𝑥𝜓
4	3	qremex-P6 723	. . 3 ⊢ (∃𝑥𝜓 ↔ 𝜓)
5	4	subimr-P3.40b.RC 328	. 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (∀𝑥𝜑 → 𝜓))
6	2, 5	subbil2-P4.RC 547	1 ⊢ ((∀𝑥𝜑 → 𝜓) ↔ ∃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: (None)