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Mirrors > Home > ILE Home > Th. List > reximia | GIF version |
Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 10-Feb-1997.) |
Ref | Expression |
---|---|
reximia.1 | ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
reximia | ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexim 2526 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) | |
2 | reximia.1 | . 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) | |
3 | 1, 2 | mprg 2489 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 ∃wrex 2417 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-4 1487 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 df-ral 2421 df-rex 2422 |
This theorem is referenced by: reximi 2529 iunpw 4401 nsmallnqq 7220 1idprl 7398 1idpru 7399 qmulz 9415 zq 9418 caubnd2 10889 sin0pilem1 12862 |
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