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Mirrors > Home > MPE Home > Th. List > Mathboxes > uunT12p1 | Structured version Visualization version GIF version |
Description: A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
uunT12p1.1 | ⊢ ((⊤ ∧ 𝜓 ∧ 𝜑) → 𝜒) |
Ref | Expression |
---|---|
uunT12p1 | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3anass 1042 | . . . 4 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜑) ↔ (⊤ ∧ (𝜓 ∧ 𝜑))) | |
2 | truan 1501 | . . . 4 ⊢ ((⊤ ∧ (𝜓 ∧ 𝜑)) ↔ (𝜓 ∧ 𝜑)) | |
3 | 1, 2 | bitri 264 | . . 3 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜑) ↔ (𝜓 ∧ 𝜑)) |
4 | ancom 466 | . . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑)) | |
5 | 3, 4 | bitr4i 267 | . 2 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜑) ↔ (𝜑 ∧ 𝜓)) |
6 | uunT12p1.1 | . 2 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜑) → 𝜒) | |
7 | 5, 6 | sylbir 225 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 ⊤wtru 1484 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 197 df-an 386 df-3an 1039 df-tru 1486 |
This theorem is referenced by: (None) |
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