Theorem eximdh 1604

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 1462	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))
4		exim 1592	. 2 ⊢ (∀𝑥(𝜓 → 𝜒) → (∃𝑥𝜓 → ∃𝑥𝜒))
5	3, 4	syl 14	1 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))

Syntax hints:   → wi 4  ∀wal 1346  ∃wex 1485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-ial 1527

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  eximd  1605  19.41h  1678  hbexd  1687  equsex  1721  equsexd  1722  spimeh  1732  sbiedh  1780  exdistrfor  1793  eximdv  1873  cbvexdh  1919  mopick2  2102  2euex  2106  bj-sbimedh  13806