Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > bianir | Structured version Visualization version GIF version |
Description: If a wff is equivalent to its conjunction with another wff, the other wwf follows. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Roger Witte, 17-Aug-2020.) |
Ref | Expression |
---|---|
bianir | ⊢ ((𝜑 ∧ (𝜓 ↔ 𝜑)) → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biimpr 209 | . 2 ⊢ ((𝜓 ↔ 𝜑) → (𝜑 → 𝜓)) | |
2 | 1 | impcom 445 | 1 ⊢ ((𝜑 ∧ (𝜓 ↔ 𝜑)) → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ 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: suppimacnv 7193 lgsqrmodndvds 24878 bnj970 30271 bnj1001 30282 bj-bibibi 31744 |
Copyright terms: Public domain | W3C validator |