Theorem rexim 3154

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 156	. . . 4 ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
2	1	ral2imi 3071	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 ¬ 𝜓 → ∀𝑥 ∈ 𝐴 ¬ 𝜑))
3	2	con3d 155	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 → ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜓))
4		dfrex2 3152	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑)
5		dfrex2 3152	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜓)
6	3, 4, 5	3imtr4g 299	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))

Syntax hints:  ¬ wn 3   → wi 4  ∀wral 3053  ∃wrex 3054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1787  df-ral 3058  df-rex 3059

This theorem is referenced by:  reximia  3156  r19.29  3167  reximdvai  3182  reximdai  3221  r19.35  3246  reupick2  4209  ss2iun  4899  chfnrn  6826  isf32lem2  9854  ptcmplem4  22806  bnj110  32409  madebdayim  33713  madebdaylemold  33721  poimirlem25  35445