Theorem eximdh 1634

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 1492	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))
4		exim 1622	. 2 ⊢ (∀𝑥(𝜓 → 𝜒) → (∃𝑥𝜓 → ∃𝑥𝜒))
5	3, 4	syl 14	1 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))

Syntax hints:   → wi 4  ∀wal 1371  ∃wex 1515

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-ial 1557

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  eximd  1635  19.41h  1708  hbexd  1717  equsex  1751  equsexd  1752  spimeh  1762  sbiedh  1810  exdistrfor  1823  eximdv  1903  cbvexdh  1950  mopick2  2137  2euex  2141  bj-sbimedh  15707