Theorem 2stdpc5 32816

Description: A double stdpc5 2076 (one direction of PM*11.3). See also 2stdpc4 2354 and 19.21vv 38575. (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 2076	. . 3 ⊢ (∀𝑦(𝜑 → 𝜓) → (𝜑 → ∀𝑦𝜓))
3	2	alimi 1739	. 2 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → ∀𝑥(𝜑 → ∀𝑦𝜓))
4		2stdpc5.1	. . 3 ⊢ Ⅎ𝑥𝜑
5	4	stdpc5 2076	. 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 1481  Ⅎwnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nf 1710

This theorem is referenced by:  ax11-pm  32819  ax11-pm2  32823