Theorem bj-19.41t 36918

Description: Closed form of 19.41 2240 from the same axioms as 19.41v 1950. The same is doable with 19.27 2232, 19.28 2233, 19.31 2239, 19.32 2238, 19.44 2242, 19.45 2243. (Contributed by BJ, 2-Dec-2023.)

Assertion

Ref	Expression
bj-19.41t	⊢ (Ⅎ'𝑥𝜓 → (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓)))

Proof of Theorem bj-19.41t

Step	Hyp	Ref	Expression
1		exancom 1862	. . 3 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜓 ∧ 𝜑))
2		bj-19.42t 36917	. . 3 ⊢ (Ⅎ'𝑥𝜓 → (∃𝑥(𝜓 ∧ 𝜑) ↔ (𝜓 ∧ ∃𝑥𝜑)))
3	1, 2	bitrid 283	. 2 ⊢ (Ⅎ'𝑥𝜓 → (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜓 ∧ ∃𝑥𝜑)))
4	3	biancomd 463	1 ⊢ (Ⅎ'𝑥𝜓 → (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∃wex 1780  Ⅎ'wnnf 36867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-bj-nnf 36868

This theorem is referenced by: (None)