Theorem qimeqallb-P6-L1 700

Description: Lemma for qimeqallb-P6 701.

Hypothesis

Ref	Expression
qimeqallb-P6-L1.1	⊢ Ⅎ𝑥𝜓

Assertion

Ref	Expression
qimeqallb-P6-L1	⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))

Proof of Theorem qimeqallb-P6-L1

Step	Hyp	Ref	Expression
1		alloverimex-P5.CL 604	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
2		qimeqallb-P6-L1.1	. . . 4 ⊢ Ⅎ𝑥𝜓
3		dfnfreealt-P6 683	. . . 4 ⊢ (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
4	2, 3	bimpf-P4.RC 532	. . 3 ⊢ (∃𝑥𝜓 → ∀𝑥𝜓)
5	4	rcp-NDIMP0addall 207	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜓 → ∀𝑥𝜓))
6	1, 5	syl-P3.24 259	1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))

Syntax hints:  ∀wff-forall 8   → wff-imp 10  ∃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:  qimeqallb-P6  701