Theorem exanOLD 1776

Description: Obsolete proof of exan 1775 as of 8-Oct-2021. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 13-Jan-2018.) Reduce axiom dependencies. (Revised by BJ, 7-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
exan.1	⊢ (∃𝑥𝜑 ∧ 𝜓)

Assertion

Ref	Expression
exanOLD	⊢ ∃𝑥(𝜑 ∧ 𝜓)

Proof of Theorem exanOLD

Step	Hyp	Ref	Expression
1		exan.1	. . 3 ⊢ (∃𝑥𝜑 ∧ 𝜓)
2	1	simpli 473	. 2 ⊢ ∃𝑥𝜑
3		pm3.21 463	. . . 4 ⊢ (𝜓 → (𝜑 → (𝜑 ∧ 𝜓)))
4	3	aleximi 1749	. . 3 ⊢ (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)))
5	1	simpri 477	. . 3 ⊢ 𝜓
6	4, 5	mpg 1715	. 2 ⊢ (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓))
7	2, 6	ax-mp 5	1 ⊢ ∃𝑥(𝜑 ∧ 𝜓)

Syntax hints:   → wi 4   ∧ wa 383  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696

This theorem is referenced by: (None)