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Mirrors > Home > MPE Home > Th. List > rexim | Structured version Visualization version 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 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)) |
Colors of variables: wff setvar class |
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 |
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