Theorem bj-19.41al 33994

Description: Special case of 19.41 2237 proved from Tarski, ax-10 2145 (modal5) and hba1 2301 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-19.41al	⊢ (∃𝑥(𝜑 ∧ ∀𝑥𝜓) ↔ (∃𝑥𝜑 ∧ ∀𝑥𝜓))

Proof of Theorem bj-19.41al

Step	Hyp	Ref	Expression
1		19.40 1887	. . 3 ⊢ (∃𝑥(𝜑 ∧ ∀𝑥𝜓) → (∃𝑥𝜑 ∧ ∃𝑥∀𝑥𝜓))
2		hbe1a 2148	. . . 4 ⊢ (∃𝑥∀𝑥𝜓 → ∀𝑥𝜓)
3	2	anim2i 618	. . 3 ⊢ ((∃𝑥𝜑 ∧ ∃𝑥∀𝑥𝜓) → (∃𝑥𝜑 ∧ ∀𝑥𝜓))
4	1, 3	syl 17	. 2 ⊢ (∃𝑥(𝜑 ∧ ∀𝑥𝜓) → (∃𝑥𝜑 ∧ ∀𝑥𝜓))
5		hba1 2301	. . . 4 ⊢ (∀𝑥𝜓 → ∀𝑥∀𝑥𝜓)
6	5	anim2i 618	. . 3 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → (∃𝑥𝜑 ∧ ∀𝑥∀𝑥𝜓))
7		19.29r 1875	. . 3 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥∀𝑥𝜓) → ∃𝑥(𝜑 ∧ ∀𝑥𝜓))
8	6, 7	syl 17	. 2 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑 ∧ ∀𝑥𝜓))
9	4, 8	impbii 211	1 ⊢ (∃𝑥(𝜑 ∧ ∀𝑥𝜓) ↔ (∃𝑥𝜑 ∧ ∀𝑥𝜓))

Syntax hints:   ↔ wb 208   ∧ wa 398  ∀wal 1535  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785

This theorem is referenced by:  bj-equsexval  33995