Theorem qimeqallbv-P5 620

Description: Second Bi-directional Form of qimeqallhalf-P5 609 ( U ↔ ( E → U ) ) (variable restriction b).

Holds when '𝑥' does not occur in '𝜓'. The most general version is qimeqallb-P6 701.

Assertion

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

Distinct variable group:   𝜓,𝑥

Proof of Theorem qimeqallbv-P5

Step	Hyp	Ref	Expression
1		qimeqallbv-P5-L1 619	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))
2		qimeqallhalf-P5 609	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))
3	1, 2	rcp-NDBII0 239	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))

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

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:  qceximlv-P5  674  psubnfr-P6  784