Theorem eximdh 1896

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 1864	. 2 ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
4	1, 3	syl 18	1 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841

This proof depends on definitions:  df-bi 210  df-ex 1812

This theorem is used by:  eximdv  1949  eximd  2252  nfeqf2  2406  bj-spim  37525  bj-spime  37526  bj-cbvalimdlem  37528  bj-cbveximdlem  37529  bj-cbveximd  37531  bj-nnf-spime  37677  ax6e2eq  45540