Theorem eximdh 1588

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 1565	. 2 ⊢ (φ → ∀x(ψ → χ))
4		exim 1575	. 2 ⊢ (∀x(ψ → χ) → (∃xψ → ∃xχ))
5	3, 4	syl 15	1 ⊢ (φ → (∃xψ → ∃xχ))

Syntax hints:   → wi 4  ∀wal 1540  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-ex 1542

This theorem is referenced by:  eximdv  1622  eximd  1770