Theorem 19.22 1038

Description: Theorem 19.22 of [Margaris] p. 90.

Assertion

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

Proof of Theorem 19.22

Step	Hyp	Ref	Expression
1		con3 94	. . . 4 ⊢ ((φ → ψ) → (¬ ψ → ¬ φ))
2	1	19.20ii 994	. . 3 ⊢ (∀x(φ → ψ) → (∀x ¬ ψ → ∀x ¬ φ))
3	2	con3d 95	. 2 ⊢ (∀x(φ → ψ) → (¬ ∀x ¬ φ → ¬ ∀x ¬ ψ))
4		df-ex 980	. 2 ⊢ (∃xφ ↔ ¬ ∀x ¬ φ)
5		df-ex 980	. 2 ⊢ (∃xψ ↔ ¬ ∀x ¬ ψ)
6	3, 4, 5	3imtr4g 552	1 ⊢ (∀x(φ → ψ) → (∃xφ → ∃xψ))

Syntax hints:  ¬ wn 2   → wi 3  ∀wal 953  ∃wex 979

This theorem is referenced by:  19.22i 1039  19.18 1049  19.22d 1061  19.23 1062  19.25 1083  ax9o 1121  sbied 1194  mo 1392  2mo 1446  r19.22 1729  chsscm 9067

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 962  ax-4 972  ax-5o 974

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980

Copyright terms: Public domain