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| Mirrors > Home > MPE Home > Th. List > anbi1ci | Structured version Visualization version GIF version | ||
| Description: Variant of anbi1i 624 with commutation. (Contributed by Peter Mazsa, 7-Mar-2020.) |
| Ref | Expression |
|---|---|
| anbi.1 | ⊢ (𝜑 ↔ 𝜓) |
| Ref | Expression |
|---|---|
| anbi1ci | ⊢ ((𝜒 ∧ 𝜑) ↔ (𝜓 ∧ 𝜒)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | anbi.1 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
| 2 | 1 | anbi2i 623 | . 2 ⊢ ((𝜒 ∧ 𝜑) ↔ (𝜒 ∧ 𝜓)) |
| 3 | 2 | biancomi 462 | 1 ⊢ ((𝜒 ∧ 𝜑) ↔ (𝜓 ∧ 𝜒)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 |
| 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 207 df-an 396 |
| This theorem is referenced by: dfid3 5581 imai 6092 frpoind 6363 wfiOLD 6372 dfac5lem3 10165 cf0 10291 eqscut2 27851 coep 35752 brtxp 35881 sscoid 35914 brapply 35939 dfrdg4 35952 wl-df4-3mintru2 37488 rnxrncnvepres 38401 rnxrnidres 38402 pmapglb 39772 polval2N 39908 rp-fakeoranass 43527 alephiso2 43571 |
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