Description: Alternate proof of bj-ssbid1 , not using sbequ1 . (Contributed by BJ, 22-Dec-2020) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | bj-ssbid1ALT | ⊢ ( 𝜑 → [ 𝑥 / 𝑥 ] 𝜑 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax12v | ⊢ ( 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) | |
| 2 | 1 | equcoms | ⊢ ( 𝑦 = 𝑥 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) |
| 3 | 2 | com12 | ⊢ ( 𝜑 → ( 𝑦 = 𝑥 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) |
| 4 | 3 | alrimiv | ⊢ ( 𝜑 → ∀ 𝑦 ( 𝑦 = 𝑥 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) |
| 5 | df-sb | ⊢ ( [ 𝑥 / 𝑥 ] 𝜑 ↔ ∀ 𝑦 ( 𝑦 = 𝑥 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) | |
| 6 | 4 5 | sylibr | ⊢ ( 𝜑 → [ 𝑥 / 𝑥 ] 𝜑 ) |