Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ad4ant23 | Structured version Visualization version GIF version |
Description: Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) |
Ref | Expression |
---|---|
ad4ant23.1 | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
Ref | Expression |
---|---|
ad4ant23 | ⊢ ((((𝜃 ∧ 𝜑) ∧ 𝜓) ∧ 𝜏) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ad4ant23.1 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | |
2 | 1 | ex 449 | . . . 4 ⊢ (𝜑 → (𝜓 → 𝜒)) |
3 | 2 | a1dd 48 | . . 3 ⊢ (𝜑 → (𝜓 → (𝜏 → 𝜒))) |
4 | 3 | a1i 11 | . 2 ⊢ (𝜃 → (𝜑 → (𝜓 → (𝜏 → 𝜒)))) |
5 | 4 | imp41 617 | 1 ⊢ ((((𝜃 ∧ 𝜑) ∧ 𝜓) ∧ 𝜏) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 |
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 |
This theorem is referenced by: fntpb 6378 matunitlindflem1 32575 matunitlindflem2 32576 heicant 32614 difmap 38394 sge0resplit 39299 hoidmvle 39490 usgredg2vlem2 40453 umgr3v3e3cycl 41351 frgrwopreglem5 41485 av-extwwlkfablem2 41510 |
Copyright terms: Public domain | W3C validator |