Theorem rexim 3093

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 3091	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 ¬ 𝜓 → ∀𝑥 ∈ 𝐴 ¬ 𝜑))
3		ralnex 3078	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜓)
4		ralnex 3078	. . 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 3067  ∃wrex 3076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-ral 3068  df-rex 3077

This theorem is referenced by:  reximiaOLD  3094  rexbi  3110  r19.35OLD  3115  r19.29OLD  3121  r19.30  3126  reximdvaiOLD  3172  reximdai  3267  reupick2  4350  ss2iun  5033  dfiun2g  5053  chfnrn  7082  isf32lem2  10423  psdmul  22193  ptcmplem4  24084  madebdayim  27944  madebdaylemold  27954  bnj110  34834  poimirlem25  37605