New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > bicom | GIF version |
Description: Commutative law for equivalence. Theorem *4.21 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
bicom | ⊢ ((φ ↔ ψ) ↔ (ψ ↔ φ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bicom1 190 | . 2 ⊢ ((φ ↔ ψ) → (ψ ↔ φ)) | |
2 | bicom1 190 | . 2 ⊢ ((ψ ↔ φ) → (φ ↔ ψ)) | |
3 | 1, 2 | impbii 180 | 1 ⊢ ((φ ↔ ψ) ↔ (ψ ↔ φ)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 |
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 177 |
This theorem is referenced by: bicomd 192 bibi1i 305 bibi1d 310 con2bi 318 ibibr 332 bibif 335 nbbn 347 pm5.17 858 biluk 899 bigolden 901 xorcom 1307 falbitru 1352 3impexpbicom 1367 mtp-xorOLD 1537 exists1 2293 eqcom 2355 abeq1 2459 ssequn1 3433 isocnv 5491 |
Copyright terms: Public domain | W3C validator |