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Mirrors > Home > MPE Home > Th. List > ancom1s | Structured version Visualization version GIF version |
Description: Inference commuting a nested conjunction in antecedent. (Contributed by NM, 24-May-2006.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) |
Ref | Expression |
---|---|
an32s.1 | ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) |
Ref | Expression |
---|---|
ancom1s | ⊢ (((𝜓 ∧ 𝜑) ∧ 𝜒) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm3.22 460 | . 2 ⊢ ((𝜓 ∧ 𝜑) → (𝜑 ∧ 𝜓)) | |
2 | an32s.1 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) | |
3 | 1, 2 | sylan 580 | 1 ⊢ (((𝜓 ∧ 𝜑) ∧ 𝜒) → 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 |
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 |
This theorem is referenced by: odi 8526 sornom 10213 leltadd 11639 divmul13 11858 absmax 15214 fzomaxdif 15228 dmatsgrp 21848 comppfsc 22883 iocopnst 24303 mumul 26530 lgsdir2 26678 branmfn 31047 chirredlem2 31333 chirredlem4 31335 icoreclin 35828 relowlssretop 35834 pibt2 35888 frinfm 36194 fzmul 36200 fdc 36204 rpnnen3 41342 |
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