Theorem bj-19.41al 37170

Description: Special case of 19.41 2277 proved from core axioms, ax-10 2182 (modal5), and hba1 2334 (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 1913	. . 3 ⊢ (∃𝑥(𝜑 ∧ ∀𝑥𝜓) → (∃𝑥𝜑 ∧ ∃𝑥∀𝑥𝜓))
2		hbe1a 2185	. . . 4 ⊢ (∃𝑥∀𝑥𝜓 → ∀𝑥𝜓)
3	2	anim2i 628	. . 3 ⊢ ((∃𝑥𝜑 ∧ ∃𝑥∀𝑥𝜓) → (∃𝑥𝜑 ∧ ∀𝑥𝜓))
4	1, 3	syl 18	. 2 ⊢ (∃𝑥(𝜑 ∧ ∀𝑥𝜓) → (∃𝑥𝜑 ∧ ∀𝑥𝜓))
5		hba1 2334	. . . 4 ⊢ (∀𝑥𝜓 → ∀𝑥∀𝑥𝜓)
6	5	anim2i 628	. . 3 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → (∃𝑥𝜑 ∧ ∀𝑥∀𝑥𝜓))
7		19.29r 1901	. . 3 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥∀𝑥𝜓) → ∃𝑥(𝜑 ∧ ∀𝑥𝜓))
8	6, 7	syl 18	. 2 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑 ∧ ∀𝑥𝜓))
9	4, 8	impbii 212	1 ⊢ (∃𝑥(𝜑 ∧ ∀𝑥𝜓) ↔ (∃𝑥𝜑 ∧ ∀𝑥𝜓))

Syntax hints:   ↔ wb 209   ∧ wa 400  ∀wal 1565  ∃wex 1806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-10 2182  ax-12 2219

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1807  df-nf 1811

This theorem is referenced by:  bj-equsexval  37171