Theorem 2stdpc5

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

	Ref	Expression
Hypotheses	2stdpc5.1	⊢ Ⅎ 𝑥 𝜑
	2stdpc5.2	⊢ Ⅎ 𝑦 𝜑
Assertion	2stdpc5	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) → ( 𝜑 → ∀ 𝑥 ∀ 𝑦 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		2stdpc5.1	⊢ Ⅎ 𝑥 𝜑
2		2stdpc5.2	⊢ Ⅎ 𝑦 𝜑
3	2	stdpc5	⊢ ( ∀ 𝑦 ( 𝜑 → 𝜓 ) → ( 𝜑 → ∀ 𝑦 𝜓 ) )
4	3	alimi	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) → ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜓 ) )
5	1	stdpc5	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜓 ) → ( 𝜑 → ∀ 𝑥 ∀ 𝑦 𝜓 ) )
6	4 5	syl	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) → ( 𝜑 → ∀ 𝑥 ∀ 𝑦 𝜓 ) )