Theorem qimeqalla-P6 699

Description: Quantified Implication Equivalence Law ( U ↔ ( E → U ) ) (non-freeness condition a).

Hypothesis

Ref	Expression
qimeqalla-P6.1	⊢ Ⅎ𝑥𝜑

Assertion

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

Proof of Theorem qimeqalla-P6

Step	Hyp	Ref	Expression
1		qimeqalla-P6.1	. . 3 ⊢ Ⅎ𝑥𝜑
2	1	qimeqalla-P6-L1 698	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))
3		qimeqallhalf-P5 609	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))
4	2, 3	rcp-NDBII0 239	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))

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

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:  qcallimr-P6  757