Theorem bj-19.41t 36797

Description: Closed form of 19.41 2236 from the same axioms as 19.41v 1949. The same is doable with 19.27 2228, 19.28 2229, 19.31 2235, 19.32 2234, 19.44 2238, 19.45 2239. (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 1861	. . 3 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜓 ∧ 𝜑))
2		bj-19.42t 36796	. . 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 1779  Ⅎ'wnnf 36746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-bj-nnf 36747

This theorem is referenced by: (None)