Theorem daraptiALT 2770

Description: Alternate proof of darapti 2769, shorter but using more axioms. See comment of dariiALT 2751. (Contributed by David A. Wheeler, 27-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
darapti.maj	⊢ ∀𝑥(𝜑 → 𝜓)
darapti.min	⊢ ∀𝑥(𝜑 → 𝜒)
darapti.e	⊢ ∃𝑥𝜑

Assertion

Ref	Expression
daraptiALT	⊢ ∃𝑥(𝜒 ∧ 𝜓)

Proof of Theorem daraptiALT

Step	Hyp	Ref	Expression
1		darapti.e	. 2 ⊢ ∃𝑥𝜑
2		darapti.min	. . . 4 ⊢ ∀𝑥(𝜑 → 𝜒)
3	2	spi 2183	. . 3 ⊢ (𝜑 → 𝜒)
4		darapti.maj	. . . 4 ⊢ ∀𝑥(𝜑 → 𝜓)
5	4	spi 2183	. . 3 ⊢ (𝜑 → 𝜓)
6	3, 5	jca 514	. 2 ⊢ (𝜑 → (𝜒 ∧ 𝜓))
7	1, 6	eximii 1837	1 ⊢ ∃𝑥(𝜒 ∧ 𝜓)

Syntax hints:   → wi 4   ∧ 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-12 2177

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781

This theorem is referenced by: (None)