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Mirrors > Home > NFE Home > Th. List > rexim | GIF version |
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.) |
Ref | Expression |
---|---|
rexim | ⊢ (∀x ∈ A (φ → ψ) → (∃x ∈ A φ → ∃x ∈ A ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | con3 126 | . . . 4 ⊢ ((φ → ψ) → (¬ ψ → ¬ φ)) | |
2 | 1 | ral2imi 2690 | . . 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 2627 | . 2 ⊢ (∃x ∈ A φ ↔ ¬ ∀x ∈ A ¬ φ) | |
5 | dfrex2 2627 | . 2 ⊢ (∃x ∈ A ψ ↔ ¬ ∀x ∈ A ¬ ψ) | |
6 | 3, 4, 5 | 3imtr4g 261 | 1 ⊢ (∀x ∈ A (φ → ψ) → (∃x ∈ A φ → ∃x ∈ A ψ)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wral 2614 ∃wrex 2615 |
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 2619 df-rex 2620 |
This theorem is referenced by: reximia 2719 reximdai 2722 r19.29 2754 reupick2 3541 ss2iun 3984 |
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