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Mirrors > Home > MPE Home > Th. List > Mathboxes > uunT1 | Structured version Visualization version GIF version |
Description: A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 3-Dec-2015.) Proof was revised to accomodate a possible future version of df-tru 1486. (Revised by David A. Wheeler, 8-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
uunT1.1 | ⊢ ((⊤ ∧ 𝜑) → 𝜓) |
Ref | Expression |
---|---|
uunT1 | ⊢ (𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orc 400 | . . 3 ⊢ (𝜑 → (𝜑 ∨ ¬ 𝜑)) | |
2 | tru 1487 | . . . . 5 ⊢ ⊤ | |
3 | biid 251 | . . . . 5 ⊢ (𝜑 ↔ 𝜑) | |
4 | 2, 3 | 2th 254 | . . . 4 ⊢ (⊤ ↔ (𝜑 ↔ 𝜑)) |
5 | exmid 431 | . . . . . 6 ⊢ (𝜑 ∨ ¬ 𝜑) | |
6 | 5 | a1i 11 | . . . . 5 ⊢ ((𝜑 ↔ 𝜑) → (𝜑 ∨ ¬ 𝜑)) |
7 | biidd 252 | . . . . 5 ⊢ ((𝜑 ∨ ¬ 𝜑) → (𝜑 ↔ 𝜑)) | |
8 | 6, 7 | impbii 199 | . . . 4 ⊢ ((𝜑 ↔ 𝜑) ↔ (𝜑 ∨ ¬ 𝜑)) |
9 | 4, 8 | bitri 264 | . . 3 ⊢ (⊤ ↔ (𝜑 ∨ ¬ 𝜑)) |
10 | 1, 9 | sylibr 224 | . 2 ⊢ (𝜑 → ⊤) |
11 | uunT1.1 | . 2 ⊢ ((⊤ ∧ 𝜑) → 𝜓) | |
12 | 10, 11 | mpancom 703 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 ⊤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-or 385 df-an 386 df-tru 1486 |
This theorem is referenced by: uunT21 39009 sspwimpALT 39161 |
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