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Mirrors > Home > MPE Home > Th. List > bianbi | Structured version Visualization version GIF version |
Description: Exchanging conjunction in a biconditional. (Contributed by Peter Mazsa, 31-Jul-2023.) |
Ref | Expression |
---|---|
bianbi.1 | ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) |
bianbi.2 | ⊢ (𝜓 ↔ 𝜃) |
Ref | Expression |
---|---|
bianbi | ⊢ (𝜑 ↔ (𝜃 ∧ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bianbi.1 | . 2 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) | |
2 | bianbi.2 | . . 3 ⊢ (𝜓 ↔ 𝜃) | |
3 | 2 | anbi1i 623 | . 2 ⊢ ((𝜓 ∧ 𝜒) ↔ (𝜃 ∧ 𝜒)) |
4 | 1, 3 | bitri 275 | 1 ⊢ (𝜑 ↔ (𝜃 ∧ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ 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 206 df-an 396 |
This theorem is referenced by: anbi12i 627 bianassc 642 pm5.53 1003 dfifp4 1065 dfifp5 1066 an3andi 1479 19.28v 1987 19.28 2217 2eu4 2646 r19.26-3 3109 3reeanv 3224 rmo4 3725 rmo3f 3729 sbc3an 3846 rmo3 3882 difin2 4292 otelxp 5722 dfrefrel5 37989 dfantisymrel4 38233 dfantisymrel5 38234 grimuhgr 47176 |
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