Theorem bj-19.42t 34123

Description: Closed form of 19.42 2237 from the same axioms as 19.42v 1953. (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 1886	. . 3 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))
2		bj-nnfe 34083	. . . 4 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → 𝜑))
3	2	anim1d 612	. . 3 ⊢ (Ⅎ'𝑥𝜑 → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) → (𝜑 ∧ ∃𝑥𝜓)))
4	1, 3	syl5 34	. 2 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) → (𝜑 ∧ ∃𝑥𝜓)))
5		bj-nnfa 34081	. . . 4 ⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
6	5	anim1d 612	. . 3 ⊢ (Ⅎ'𝑥𝜑 → ((𝜑 ∧ ∃𝑥𝜓) → (∀𝑥𝜑 ∧ ∃𝑥𝜓)))
7		19.29 1873	. . 3 ⊢ ((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓))
8	6, 7	syl6 35	. 2 ⊢ (Ⅎ'𝑥𝜑 → ((𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓)))
9	4, 8	impbid 214	1 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1534  ∃wex 1779  Ⅎ'wnnf 34076

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 209  df-an 399  df-ex 1780  df-bj-nnf 34077

This theorem is referenced by:  bj-19.41t  34124