Theorem bj-19.41t 37486

Description: Closed form of 19.41 2273 from the same axioms as 19.41v 1982. The same is doable with 19.27 2265, 19.28 2266, 19.31 2272, 19.32 2271, 19.44 2275, 19.45 2276. (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 1894	. . 3 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜓 ∧ 𝜑))
2		bj-19.42t 37485	. . 3 ⊢ (Ⅎ'𝑥𝜓 → (∃𝑥(𝜓 ∧ 𝜑) ↔ (𝜓 ∧ ∃𝑥𝜑)))
3	1, 2	bitrid 286	. 2 ⊢ (Ⅎ'𝑥𝜓 → (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜓 ∧ ∃𝑥𝜑)))
4	3	biancomd 469	1 ⊢ (Ⅎ'𝑥𝜓 → (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∃wex 1812  Ⅎ'wnnf 37446

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-bj-nnf 37447

This theorem is used by: (None)