Theorem 2stdpc5 37833

Description: A double stdpc5 2244 (one direction of PM*11.3). See also 2stdpc4 2106 and 19.21vv 45453. (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 2244	. . 3 ⊢ (∀𝑦(𝜑 → 𝜓) → (𝜑 → ∀𝑦𝜓))
3	2	alimi 1843	. 2 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → ∀𝑥(𝜑 → ∀𝑦𝜓))
4		2stdpc5.1	. . 3 ⊢ Ⅎ𝑥𝜑
5	4	stdpc5 2244	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜓) → (𝜑 → ∀𝑥∀𝑦𝜓))
6	3, 5	syl 18	1 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (𝜑 → ∀𝑥∀𝑦𝜓))

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  Ⅎwnf 1815

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-12 2212

This proof depends on definitions:  df-bi 210  df-ex 1812  df-nf 1816

This theorem is used by:  ax11-pm  37836  ax11-pm2  37840