Theorem bj-19.41t 37242

Description: Closed form of 19.41 2271 from the same axioms as 19.41v 1970. The same is doable with 19.27 2263, 19.28 2264, 19.31 2270, 19.32 2269, 19.44 2273, 19.45 2274. (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 1882	. . 3 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜓 ∧ 𝜑))
2		bj-19.42t 37241	. . 3 ⊢ (Ⅎ'𝑥𝜓 → (∃𝑥(𝜓 ∧ 𝜑) ↔ (𝜓 ∧ ∃𝑥𝜑)))
3	1, 2	bitrid 285	. 2 ⊢ (Ⅎ'𝑥𝜓 → (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜓 ∧ ∃𝑥𝜑)))
4	3	biancomd 467	1 ⊢ (Ⅎ'𝑥𝜓 → (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∃wex 1800  Ⅎ'wnnf 37202

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1801  df-bj-nnf 37203

This theorem is referenced by: (None)