Theorem barbariALT 2669

Description: Alternate proof of barbari 2668, shorter but using more axioms. See comment of dariiALT 2665. (Contributed by David A. Wheeler, 27-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
barbariALT	⊢ ∃𝑥(𝜒 ∧ 𝜓)

Proof of Theorem barbariALT

Step	Hyp	Ref	Expression
1		barbari.e	. 2 ⊢ ∃𝑥𝜒
2		barbari.maj	. . . . 5 ⊢ ∀𝑥(𝜑 → 𝜓)
3		barbari.min	. . . . 5 ⊢ ∀𝑥(𝜒 → 𝜑)
4	2, 3	barbara 2662	. . . 4 ⊢ ∀𝑥(𝜒 → 𝜓)
5	4	spi 2183	. . 3 ⊢ (𝜒 → 𝜓)
6	5	ancli 548	. 2 ⊢ (𝜒 → (𝜒 ∧ 𝜓))
7	1, 6	eximii 1836	1 ⊢ ∃𝑥(𝜒 ∧ 𝜓)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1537  ∃wex 1778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-12 2176

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779

This theorem is referenced by: (None)