Theorem bj-19.42t 34217

Description: Closed form of 19.42 2236 from the same axioms as 19.42v 1954. (Contributed by BJ, 2-Dec-2023.)

Assertion

Ref	Expression
bj-19.42t	⊢ (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓)))

Proof of Theorem bj-19.42t

Step	Hyp	Ref	Expression
1		19.40 1887	. . 3 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))
2		bj-nnfe 34177	. . . 4 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → 𝜑))
3	2	anim1d 613	. . 3 ⊢ (Ⅎ'𝑥𝜑 → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) → (𝜑 ∧ ∃𝑥𝜓)))
4	1, 3	syl5 34	. 2 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) → (𝜑 ∧ ∃𝑥𝜓)))
5		bj-nnfa 34175	. . . 4 ⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
6	5	anim1d 613	. . 3 ⊢ (Ⅎ'𝑥𝜑 → ((𝜑 ∧ ∃𝑥𝜓) → (∀𝑥𝜑 ∧ ∃𝑥𝜓)))
7		19.29 1874	. . 3 ⊢ ((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓))
8	6, 7	syl6 35	. 2 ⊢ (Ⅎ'𝑥𝜑 → ((𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓)))
9	4, 8	impbid 215	1 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781  Ⅎ'wnnf 34170

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 210  df-an 400  df-ex 1782  df-bj-nnf 34171

This theorem is referenced by:  bj-19.41t  34218