Theorem bj-19.21t 33644

Description: Proof of 19.21t 2135 from stdpc5t 33641. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-19.21t	⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)))

Proof of Theorem bj-19.21t

Step	Hyp	Ref	Expression
1		stdpc5t 33641	. 2 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))
2		19.9t 2133	. . . 4 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑))
3	2	imbi1d 334	. . 3 ⊢ (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
4		19.38 1801	. . 3 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))
5	3, 4	syl6bir 246	. 2 ⊢ (Ⅎ𝑥𝜑 → ((𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓)))
6	1, 5	impbid 204	1 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)))

Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1505  ∃wex 1742  Ⅎwnf 1746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-12 2106

This theorem depends on definitions:  df-bi 199  df-ex 1743  df-nf 1747

This theorem is referenced by: (None)