Theorem exim 1575

Description: Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)

Assertion

Ref	Expression
exim	⊢ (∀x(φ → ψ) → (∃xφ → ∃xψ))

Proof of Theorem exim

Step	Hyp	Ref	Expression
1		con3 126	. . . 4 ⊢ ((φ → ψ) → (¬ ψ → ¬ φ))
2	1	al2imi 1561	. . 3 ⊢ (∀x(φ → ψ) → (∀x ¬ ψ → ∀x ¬ φ))
3		alnex 1543	. . 3 ⊢ (∀x ¬ ψ ↔ ¬ ∃xψ)
4		alnex 1543	. . 3 ⊢ (∀x ¬ φ ↔ ¬ ∃xφ)
5	2, 3, 4	3imtr3g 260	. 2 ⊢ (∀x(φ → ψ) → (¬ ∃xψ → ¬ ∃xφ))
6	5	con4d 97	1 ⊢ (∀x(φ → ψ) → (∃xφ → ∃xψ))

Syntax hints:  ¬ wn 3   → 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:  eximi  1576  exbi  1581  eximdh  1588  19.29  1596  19.25  1603  19.30  1604  19.23t  1800  19.23tOLD  1819  19.23hOLD  1820  2mo  2282  elex22  2871  elex2  2872  vtoclegft  2927  spcimgft  2931