Theorem qceximlv-P5 674

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

'𝑥' cannot occur in '𝜓'.

The most general form is qceximl-P6 760.

Assertion

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

Distinct variable group:   𝜓,𝑥

Proof of Theorem qceximlv-P5

Step	Hyp	Ref	Expression
1		qimeqallbv-P5 620	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))
2	1	bisym-P3.33b.RC 299	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓))
3		qremallv-P5 656	. . 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