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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 2558 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) | |
2 | reximia.1 | . 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) | |
3 | 1, 2 | mprg 2521 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2135 ∃wrex 2443 |
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 1434 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-4 1497 ax-ial 1521 |
This theorem depends on definitions: df-bi 116 df-ral 2447 df-rex 2448 |
This theorem is referenced by: reximi 2561 iunpw 4452 nsmallnqq 7344 1idprl 7522 1idpru 7523 qmulz 9552 zq 9555 caubnd2 11045 sin0pilem1 13243 |
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