Theorem qcallimlv-P5 673

Description: Quantifier Collection Law: Universal Quantifier Left on Implication (variable restriction).

'𝑥' cannot occur in '𝜓'.

The most general form is qcalliml-P6 759.

Assertion

Ref	Expression
qcallimlv-P5	⊢ ((∀𝑥𝜑 → 𝜓) ↔ ∃𝑥(𝜑 → 𝜓))

Distinct variable group:   𝜓,𝑥

Proof of Theorem qcallimlv-P5

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		qremexv-P5 657	. . 3 ⊢ (∃𝑥𝜓 ↔ 𝜓)
4	3	subimr-P3.40b.RC 328	. 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (∀𝑥𝜑 → 𝜓))
5	2, 4	subbil2-P4.RC 547	1 ⊢ ((∀𝑥𝜑 → 𝜓) ↔ ∃𝑥(𝜑 → 𝜓))

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

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

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

This theorem is referenced by:  example-E5.04a  675