Theorem bj-19.41t 37013

Description: Closed form of 19.41 2243 from the same axioms as 19.41v 1951. The same is doable with 19.27 2235, 19.28 2236, 19.31 2242, 19.32 2241, 19.44 2245, 19.45 2246. (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 1863	. . 3 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜓 ∧ 𝜑))
2		bj-19.42t 37012	. . 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 1781  Ⅎ'wnnf 36973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-bj-nnf 36974

This theorem is referenced by: (None)