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Mirrors > Home > MPE Home > Th. List > Mathboxes > e12an | Structured version Visualization version GIF version |
Description: Conjunction form of e12 42304 (see syl6an 681). (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
e12an.1 | ⊢ ( 𝜑 ▶ 𝜓 ) |
e12an.2 | ⊢ ( 𝜑 , 𝜒 ▶ 𝜃 ) |
e12an.3 | ⊢ ((𝜓 ∧ 𝜃) → 𝜏) |
Ref | Expression |
---|---|
e12an | ⊢ ( 𝜑 , 𝜒 ▶ 𝜏 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | e12an.1 | . 2 ⊢ ( 𝜑 ▶ 𝜓 ) | |
2 | e12an.2 | . 2 ⊢ ( 𝜑 , 𝜒 ▶ 𝜃 ) | |
3 | e12an.3 | . . 3 ⊢ ((𝜓 ∧ 𝜃) → 𝜏) | |
4 | 3 | ex 413 | . 2 ⊢ (𝜓 → (𝜃 → 𝜏)) |
5 | 1, 2, 4 | e12 42304 | 1 ⊢ ( 𝜑 , 𝜒 ▶ 𝜏 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ( wvd1 42149 ( wvd2 42157 |
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-an 397 df-vd1 42150 df-vd2 42158 |
This theorem is referenced by: sstrALT2VD 42414 |
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