Theorem stdpc5t 37833

Description: Closed form of stdpc5 2244. (Possible to place it before 19.21t 2242 and use it to prove 19.21t 2242). (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)

Assertion

Ref	Expression
stdpc5t	⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))

Proof of Theorem stdpc5t

Step	Hyp	Ref	Expression
1		nf5r 2230	. 2 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
2		alim 1842	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
3	1, 2	syl9 78	1 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))

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:  bj-stdpc5  37834  bj-19.21t0  37836