Theorem 2stdpc5 36214

Description: A double stdpc5 2193 (one direction of PM*11.3). See also 2stdpc4 2065 and 19.21vv 43693. (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 2193	. . 3 ⊢ (∀𝑦(𝜑 → 𝜓) → (𝜑 → ∀𝑦𝜓))
3	2	alimi 1805	. 2 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → ∀𝑥(𝜑 → ∀𝑦𝜓))
4		2stdpc5.1	. . 3 ⊢ Ⅎ𝑥𝜑
5	4	stdpc5 2193	. 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 1531  Ⅎwnf 1777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-12 2163

This theorem depends on definitions:  df-bi 206  df-ex 1774  df-nf 1778

This theorem is referenced by:  ax11-pm  36217  ax11-pm2  36221