Step |
Hyp |
Ref |
Expression |
1 |
|
excxor |
⊢ ( ( 𝜑 ⊻ 𝜓 ) ↔ ( ( 𝜑 ∧ ¬ 𝜓 ) ∨ ( ¬ 𝜑 ∧ 𝜓 ) ) ) |
2 |
1
|
orbi2i |
⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜑 ⊻ 𝜓 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ( ( 𝜑 ∧ ¬ 𝜓 ) ∨ ( ¬ 𝜑 ∧ 𝜓 ) ) ) ) |
3 |
|
orc |
⊢ ( 𝜑 → ( 𝜑 ∨ 𝜓 ) ) |
4 |
|
exmid |
⊢ ( 𝜓 ∨ ¬ 𝜓 ) |
5 |
|
pm3.2 |
⊢ ( 𝜑 → ( 𝜓 → ( 𝜑 ∧ 𝜓 ) ) ) |
6 |
|
biimp |
⊢ ( ( 𝜑 ↔ 𝜓 ) → ( 𝜑 → 𝜓 ) ) |
7 |
|
iman |
⊢ ( ( 𝜑 → 𝜓 ) ↔ ¬ ( 𝜑 ∧ ¬ 𝜓 ) ) |
8 |
6 7
|
sylib |
⊢ ( ( 𝜑 ↔ 𝜓 ) → ¬ ( 𝜑 ∧ ¬ 𝜓 ) ) |
9 |
8
|
con2i |
⊢ ( ( 𝜑 ∧ ¬ 𝜓 ) → ¬ ( 𝜑 ↔ 𝜓 ) ) |
10 |
9
|
ex |
⊢ ( 𝜑 → ( ¬ 𝜓 → ¬ ( 𝜑 ↔ 𝜓 ) ) ) |
11 |
|
df-xor |
⊢ ( ( 𝜑 ⊻ 𝜓 ) ↔ ¬ ( 𝜑 ↔ 𝜓 ) ) |
12 |
11
|
bicomi |
⊢ ( ¬ ( 𝜑 ↔ 𝜓 ) ↔ ( 𝜑 ⊻ 𝜓 ) ) |
13 |
10 12
|
syl6ib |
⊢ ( 𝜑 → ( ¬ 𝜓 → ( 𝜑 ⊻ 𝜓 ) ) ) |
14 |
5 13
|
orim12d |
⊢ ( 𝜑 → ( ( 𝜓 ∨ ¬ 𝜓 ) → ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜑 ⊻ 𝜓 ) ) ) ) |
15 |
4 14
|
mpi |
⊢ ( 𝜑 → ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜑 ⊻ 𝜓 ) ) ) |
16 |
3 15
|
2thd |
⊢ ( 𝜑 → ( ( 𝜑 ∨ 𝜓 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜑 ⊻ 𝜓 ) ) ) ) |
17 |
|
bicom |
⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( 𝜓 ↔ 𝜑 ) ) |
18 |
|
bibif |
⊢ ( ¬ 𝜑 → ( ( 𝜓 ↔ 𝜑 ) ↔ ¬ 𝜓 ) ) |
19 |
17 18
|
syl5bb |
⊢ ( ¬ 𝜑 → ( ( 𝜑 ↔ 𝜓 ) ↔ ¬ 𝜓 ) ) |
20 |
19
|
con2bid |
⊢ ( ¬ 𝜑 → ( 𝜓 ↔ ¬ ( 𝜑 ↔ 𝜓 ) ) ) |
21 |
20 12
|
syl6bb |
⊢ ( ¬ 𝜑 → ( 𝜓 ↔ ( 𝜑 ⊻ 𝜓 ) ) ) |
22 |
|
biorf |
⊢ ( ¬ 𝜑 → ( 𝜓 ↔ ( 𝜑 ∨ 𝜓 ) ) ) |
23 |
|
simpl |
⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜑 ) |
24 |
|
biorf |
⊢ ( ¬ ( 𝜑 ∧ 𝜓 ) → ( ( 𝜑 ⊻ 𝜓 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜑 ⊻ 𝜓 ) ) ) ) |
25 |
23 24
|
nsyl5 |
⊢ ( ¬ 𝜑 → ( ( 𝜑 ⊻ 𝜓 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜑 ⊻ 𝜓 ) ) ) ) |
26 |
21 22 25
|
3bitr3d |
⊢ ( ¬ 𝜑 → ( ( 𝜑 ∨ 𝜓 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜑 ⊻ 𝜓 ) ) ) ) |
27 |
16 26
|
pm2.61i |
⊢ ( ( 𝜑 ∨ 𝜓 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜑 ⊻ 𝜓 ) ) ) |
28 |
|
3orass |
⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜑 ∧ ¬ 𝜓 ) ∨ ( ¬ 𝜑 ∧ 𝜓 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ( ( 𝜑 ∧ ¬ 𝜓 ) ∨ ( ¬ 𝜑 ∧ 𝜓 ) ) ) ) |
29 |
2 27 28
|
3bitr4i |
⊢ ( ( 𝜑 ∨ 𝜓 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜑 ∧ ¬ 𝜓 ) ∨ ( ¬ 𝜑 ∧ 𝜓 ) ) ) |