Step |
Hyp |
Ref |
Expression |
1 |
|
bnj156.1 |
⊢ ( 𝜁0 ↔ ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ) |
2 |
|
bnj156.2 |
⊢ ( 𝜁1 ↔ [ 𝑔 / 𝑓 ] 𝜁0 ) |
3 |
|
bnj156.3 |
⊢ ( 𝜑1 ↔ [ 𝑔 / 𝑓 ] 𝜑′ ) |
4 |
|
bnj156.4 |
⊢ ( 𝜓1 ↔ [ 𝑔 / 𝑓 ] 𝜓′ ) |
5 |
1
|
sbcbii |
⊢ ( [ 𝑔 / 𝑓 ] 𝜁0 ↔ [ 𝑔 / 𝑓 ] ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ) |
6 |
|
sbc3an |
⊢ ( [ 𝑔 / 𝑓 ] ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ↔ ( [ 𝑔 / 𝑓 ] 𝑓 Fn 1o ∧ [ 𝑔 / 𝑓 ] 𝜑′ ∧ [ 𝑔 / 𝑓 ] 𝜓′ ) ) |
7 |
|
bnj62 |
⊢ ( [ 𝑔 / 𝑓 ] 𝑓 Fn 1o ↔ 𝑔 Fn 1o ) |
8 |
3
|
bicomi |
⊢ ( [ 𝑔 / 𝑓 ] 𝜑′ ↔ 𝜑1 ) |
9 |
4
|
bicomi |
⊢ ( [ 𝑔 / 𝑓 ] 𝜓′ ↔ 𝜓1 ) |
10 |
7 8 9
|
3anbi123i |
⊢ ( ( [ 𝑔 / 𝑓 ] 𝑓 Fn 1o ∧ [ 𝑔 / 𝑓 ] 𝜑′ ∧ [ 𝑔 / 𝑓 ] 𝜓′ ) ↔ ( 𝑔 Fn 1o ∧ 𝜑1 ∧ 𝜓1 ) ) |
11 |
6 10
|
bitri |
⊢ ( [ 𝑔 / 𝑓 ] ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ↔ ( 𝑔 Fn 1o ∧ 𝜑1 ∧ 𝜓1 ) ) |
12 |
5 11
|
bitri |
⊢ ( [ 𝑔 / 𝑓 ] 𝜁0 ↔ ( 𝑔 Fn 1o ∧ 𝜑1 ∧ 𝜓1 ) ) |
13 |
2 12
|
bitri |
⊢ ( 𝜁1 ↔ ( 𝑔 Fn 1o ∧ 𝜑1 ∧ 𝜓1 ) ) |