Theorem 19.38bOLD 1938

Description: Obsolete version of 19.38b 1937 as of 9-Jul-2022. (Contributed by BJ, 3-Nov-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
19.38bOLD	⊢ (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓)))

Proof of Theorem 19.38bOLD

Step	Hyp	Ref	Expression
1		19.38 1934	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))
2		df-nf 1880	. . 3 ⊢ (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
3		exim 1929	. . . 4 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
4		imim2 58	. . . 4 ⊢ ((∃𝑥𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑 → ∃𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
5	3, 4	syl5 34	. . 3 ⊢ ((∃𝑥𝜓 → ∀𝑥𝜓) → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
6	2, 5	sylbi 209	. 2 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
7	1, 6	impbid2 218	1 ⊢ (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1651  ∃wex 1875  Ⅎwnf 1879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905

This theorem depends on definitions:  df-bi 199  df-ex 1876  df-nf 1880

This theorem is referenced by: (None)