Theorem 2stdpc5 36830

Description: A double stdpc5 2208 (one direction of PM*11.3). See also 2stdpc4 2070 and 19.21vv 44395. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
2stdpc5.1	⊢ Ⅎ𝑥𝜑
2stdpc5.2	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
2stdpc5	⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (𝜑 → ∀𝑥∀𝑦𝜓))

Proof of Theorem 2stdpc5

Step	Hyp	Ref	Expression
1		2stdpc5.2	. . . 4 ⊢ Ⅎ𝑦𝜑
2	1	stdpc5 2208	. . 3 ⊢ (∀𝑦(𝜑 → 𝜓) → (𝜑 → ∀𝑦𝜓))
3	2	alimi 1811	. 2 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → ∀𝑥(𝜑 → ∀𝑦𝜓))
4		2stdpc5.1	. . 3 ⊢ Ⅎ𝑥𝜑
5	4	stdpc5 2208	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜓) → (𝜑 → ∀𝑥∀𝑦𝜓))
6	3, 5	syl 17	1 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (𝜑 → ∀𝑥∀𝑦𝜓))

Syntax hints:   → wi 4  ∀wal 1538  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-12 2177

This theorem depends on definitions:  df-bi 207  df-ex 1780  df-nf 1784

This theorem is referenced by:  ax11-pm  36833  ax11-pm2  36837