Theorem eximdh 1625

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	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
2		eximdh.2	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	1, 2	alrimih 1483	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))
4		exim 1613	. 2 ⊢ (∀𝑥(𝜓 → 𝜒) → (∃𝑥𝜓 → ∃𝑥𝜒))
5	3, 4	syl 14	1 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))

Syntax hints:   → wi 4  ∀wal 1362  ∃wex 1506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-ial 1548

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  eximd  1626  19.41h  1699  hbexd  1708  equsex  1742  equsexd  1743  spimeh  1753  sbiedh  1801  exdistrfor  1814  eximdv  1894  cbvexdh  1941  mopick2  2128  2euex  2132  bj-sbimedh  15417