Theorem eximdh 1867

Description: Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 20-May-1996.)

Hypotheses

Ref	Expression
eximdh.1	⊢ (𝜑 → ∀𝑥𝜑)
eximdh.2	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
eximdh	⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))

Proof of Theorem eximdh

Step	Hyp	Ref	Expression
1		eximdh.1	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2		eximdh.2	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	aleximi 1834	. 2 ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
4	1, 3	syl 17	1 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))

Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812

This theorem depends on definitions:  df-bi 206  df-ex 1783

This theorem is referenced by:  eximdv  1920  eximd  2209  nfeqf2  2377  ax6e2eq  42177