Theorem 19.22 722

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 86	. . . 4 ⊢ ((φ → ψ) → (¬ ψ → ¬ φ))
2	1	19.20ii 692	. . 3 ⊢ (∀x(φ → ψ) → (∀x ¬ ψ → ∀x ¬ φ))
3	2	con3d 87	. 2 ⊢ (∀x(φ → ψ) → (¬ ∀x ¬ φ → ¬ ∀x ¬ ψ))
4		df-ex 679	. 2 ⊢ (∃xφ ↔ ¬ ∀x ¬ φ)
5		df-ex 679	. 2 ⊢ (∃xψ ↔ ¬ ∀x ¬ ψ)
6	3, 4, 5	3imtr4g 426	1 ⊢ (∀x(φ → ψ) → (∃xφ → ∃xψ))

Syntax hints:  ¬ wn 1   → wi 2  ∀wal 672  ∃wex 678

This theorem is referenced by:  19.22i 723  19.18 732  19.22d 744  19.23 745  19.25 763  ax9 807  sbied 903  mo 1020  r19.22 1272  chsscm 5147

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679

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