Theorem rexim 2719

Description: Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Nov-1994.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Assertion

Ref	Expression
rexim	⊢ (∀x ∈ A (φ → ψ) → (∃x ∈ A φ → ∃x ∈ A ψ))

Proof of Theorem rexim

Step	Hyp	Ref	Expression
1		con3 126	. . . 4 ⊢ ((φ → ψ) → (¬ ψ → ¬ φ))
2	1	ral2imi 2691	. . 3 ⊢ (∀x ∈ A (φ → ψ) → (∀x ∈ A ¬ ψ → ∀x ∈ A ¬ φ))
3	2	con3d 125	. 2 ⊢ (∀x ∈ A (φ → ψ) → (¬ ∀x ∈ A ¬ φ → ¬ ∀x ∈ A ¬ ψ))
4		dfrex2 2628	. 2 ⊢ (∃x ∈ A φ ↔ ¬ ∀x ∈ A ¬ φ)
5		dfrex2 2628	. 2 ⊢ (∃x ∈ A ψ ↔ ¬ ∀x ∈ A ¬ ψ)
6	3, 4, 5	3imtr4g 261	1 ⊢ (∀x ∈ A (φ → ψ) → (∃x ∈ A φ → ∃x ∈ A ψ))

Syntax hints:  ¬ wn 3   → wi 4  ∀wral 2615  ∃wrex 2616

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-an 360  df-ex 1542  df-ral 2620  df-rex 2621

This theorem is referenced by:  reximia  2720  reximdai  2723  r19.29  2755  reupick2  3542  ss2iun  3985