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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 3156 | . . 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 3239 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑) | |
5 | dfrex2 3239 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜓) | |
6 | 3, 4, 5 | 3imtr4g 298 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wral 3138 ∃wrex 3139 |
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 209 df-an 399 df-ex 1781 df-ral 3143 df-rex 3144 |
This theorem is referenced by: reximia 3242 r19.29 3254 reximdvai 3272 reximdai 3311 r19.35 3341 reupick2 4289 ss2iun 4937 chfnrn 6819 isf32lem2 9776 ptcmplem4 22663 bnj110 32130 poimirlem25 34932 |
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