Theorem qimeqallbv-P5-L1 619

Description: Lemma for qimeqallbv-P5 620.

Assertion

Ref	Expression
qimeqallbv-P5-L1	⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))

Distinct variable group:   𝜓,𝑥

Proof of Theorem qimeqallbv-P5-L1

Step	Hyp	Ref	Expression
1		rcp-NDASM1of1 192	. . 3 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜓))
2	1	dalloverimex-P5.RC.GEN 607	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
3		axL5ex-P5 613	. . 3 ⊢ (∃𝑥𝜓 → 𝜓)
4	3	rcp-NDIMP0addall 207	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜓 → 𝜓))
5		ax-L5 17	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
6	5	rcp-NDIMP0addall 207	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜓 → ∀𝑥𝜓))
7	2, 4, 6	dsyl-P3.25 261	1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))

Syntax hints:  ∀wff-forall 8   → wff-imp 10  ∃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:  qimeqallbv-P5  620