Step |
Hyp |
Ref |
Expression |
1 |
|
frege56c.b |
⊢ 𝐵 ∈ 𝐶 |
2 |
1
|
frege54cor1c |
⊢ [ 𝐵 / 𝑥 ] 𝑥 = 𝐵 |
3 |
|
frege53c |
⊢ ( [ 𝐵 / 𝑥 ] 𝑥 = 𝐵 → ( 𝐵 = 𝐴 → [ 𝐴 / 𝑥 ] 𝑥 = 𝐵 ) ) |
4 |
2 3
|
ax-mp |
⊢ ( 𝐵 = 𝐴 → [ 𝐴 / 𝑥 ] 𝑥 = 𝐵 ) |
5 |
|
frege55lem1c |
⊢ ( ( 𝐵 = 𝐴 → [ 𝐴 / 𝑥 ] 𝑥 = 𝐵 ) → ( 𝐵 = 𝐴 → 𝐴 = 𝐵 ) ) |
6 |
4 5
|
ax-mp |
⊢ ( 𝐵 = 𝐴 → 𝐴 = 𝐵 ) |
7 |
|
frege9 |
⊢ ( ( 𝐵 = 𝐴 → 𝐴 = 𝐵 ) → ( ( 𝐴 = 𝐵 → ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐵 / 𝑥 ] 𝜑 ) ) → ( 𝐵 = 𝐴 → ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐵 / 𝑥 ] 𝜑 ) ) ) ) |
8 |
6 7
|
ax-mp |
⊢ ( ( 𝐴 = 𝐵 → ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐵 / 𝑥 ] 𝜑 ) ) → ( 𝐵 = 𝐴 → ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐵 / 𝑥 ] 𝜑 ) ) ) |