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Mirrors > Home > MPE Home > Th. List > Mathboxes > e1111 | Structured version Visualization version GIF version |
Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 6-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
e1111.1 | ⊢ ( 𝜑 ▶ 𝜓 ) |
e1111.2 | ⊢ ( 𝜑 ▶ 𝜒 ) |
e1111.3 | ⊢ ( 𝜑 ▶ 𝜃 ) |
e1111.4 | ⊢ ( 𝜑 ▶ 𝜏 ) |
e1111.5 | ⊢ (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂)))) |
Ref | Expression |
---|---|
e1111 | ⊢ ( 𝜑 ▶ 𝜂 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | e1111.1 | . . . 4 ⊢ ( 𝜑 ▶ 𝜓 ) | |
2 | 1 | in1 42161 | . . 3 ⊢ (𝜑 → 𝜓) |
3 | e1111.2 | . . . 4 ⊢ ( 𝜑 ▶ 𝜒 ) | |
4 | 3 | in1 42161 | . . 3 ⊢ (𝜑 → 𝜒) |
5 | e1111.3 | . . . 4 ⊢ ( 𝜑 ▶ 𝜃 ) | |
6 | 5 | in1 42161 | . . 3 ⊢ (𝜑 → 𝜃) |
7 | e1111.4 | . . . 4 ⊢ ( 𝜑 ▶ 𝜏 ) | |
8 | 7 | in1 42161 | . . 3 ⊢ (𝜑 → 𝜏) |
9 | e1111.5 | . . 3 ⊢ (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂)))) | |
10 | 2, 4, 6, 8, 9 | ee1111 42106 | . 2 ⊢ (𝜑 → 𝜂) |
11 | 10 | dfvd1ir 42163 | 1 ⊢ ( 𝜑 ▶ 𝜂 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ( wvd1 42159 |
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 206 df-vd1 42160 |
This theorem is referenced by: trsbcVD 42467 |
Copyright terms: Public domain | W3C validator |