Theorem rexim 3074

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 153	. . . 4 ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
2	1	ral2imi 3072	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 ¬ 𝜓 → ∀𝑥 ∈ 𝐴 ¬ 𝜑))
3		ralnex 3059	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜓)
4		ralnex 3059	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑)
5	2, 3, 4	3imtr3g 295	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (¬ ∃𝑥 ∈ 𝐴 𝜓 → ¬ ∃𝑥 ∈ 𝐴 𝜑))
6	5	con4d 115	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))

Syntax hints:  ¬ wn 3   → wi 4  ∀wral 3048  ∃wrex 3057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-ral 3049  df-rex 3058

This theorem is referenced by:  rexbi  3089  r19.30  3100  reximdai  3235  reupick2  4280  ss2iun  4960  dfiun2g  4980  chfnrn  6988  isf32lem2  10252  psdmul  22082  ptcmplem4  23971  madebdayim  27834  madebdaylemold  27844  bnj110  34891  poimirlem25  37705