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Mirrors > Home > MPE Home > Th. List > Mathboxes > uun2221 | Structured version Visualization version GIF version |
Description: A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 30-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
uun2221.1 | ⊢ ((𝜑 ∧ 𝜑 ∧ (𝜓 ∧ 𝜑)) → 𝜒) |
Ref | Expression |
---|---|
uun2221 | ⊢ ((𝜓 ∧ 𝜑) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uun2221.1 | . 2 ⊢ ((𝜑 ∧ 𝜑 ∧ (𝜓 ∧ 𝜑)) → 𝜒) | |
2 | 3anass 1035 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜑 ∧ (𝜓 ∧ 𝜑)) ↔ (𝜑 ∧ (𝜑 ∧ (𝜓 ∧ 𝜑)))) | |
3 | anabs5 847 | . . . . . 6 ⊢ ((𝜑 ∧ (𝜑 ∧ (𝜓 ∧ 𝜑))) ↔ (𝜑 ∧ (𝜓 ∧ 𝜑))) | |
4 | 2, 3 | bitri 263 | . . . . 5 ⊢ ((𝜑 ∧ 𝜑 ∧ (𝜓 ∧ 𝜑)) ↔ (𝜑 ∧ (𝜓 ∧ 𝜑))) |
5 | ancom 465 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑)) | |
6 | 5 | anbi2i 726 | . . . . 5 ⊢ ((𝜑 ∧ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ (𝜓 ∧ 𝜑))) |
7 | 4, 6 | bitr4i 266 | . . . 4 ⊢ ((𝜑 ∧ 𝜑 ∧ (𝜓 ∧ 𝜑)) ↔ (𝜑 ∧ (𝜑 ∧ 𝜓))) |
8 | anabs5 847 | . . . . 5 ⊢ ((𝜑 ∧ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ 𝜓)) | |
9 | 8, 5 | bitri 263 | . . . 4 ⊢ ((𝜑 ∧ (𝜑 ∧ 𝜓)) ↔ (𝜓 ∧ 𝜑)) |
10 | 7, 9 | bitri 263 | . . 3 ⊢ ((𝜑 ∧ 𝜑 ∧ (𝜓 ∧ 𝜑)) ↔ (𝜓 ∧ 𝜑)) |
11 | 10 | imbi1i 338 | . 2 ⊢ (((𝜑 ∧ 𝜑 ∧ (𝜓 ∧ 𝜑)) → 𝜒) ↔ ((𝜓 ∧ 𝜑) → 𝜒)) |
12 | 1, 11 | mpbi 219 | 1 ⊢ ((𝜓 ∧ 𝜑) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 |
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 196 df-an 385 df-3an 1033 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |