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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 2461 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) | |
2 | reximia.1 | . 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) | |
3 | 1, 2 | mprg 2426 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1434 ∃wrex 2354 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1377 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-4 1441 ax-ial 1468 |
This theorem depends on definitions: df-bi 115 df-ral 2358 df-rex 2359 |
This theorem is referenced by: reximi 2464 iunpw 4265 nsmallnqq 6874 1idprl 7052 1idpru 7053 qmulz 9003 zq 9006 caubnd2 10377 |
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