Description: Obsolete version of sbi1 as of 24-Jul-2023. Move implication out of substitution. Version of sbi1 with a disjoint variable condition, not requiring ax-13 . (Contributed by Wolf Lammen, 18-Jan-2023) (New usage is discouraged.) (Proof modification is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | sbi1vOLD | ⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) → ( [ 𝑦 / 𝑥 ] 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sb4vOLD | ⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) ) | |
| 2 | sb4vOLD | ⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) | |
| 3 | ax-2 | ⊢ ( ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) → ( ( 𝑥 = 𝑦 → 𝜑 ) → ( 𝑥 = 𝑦 → 𝜓 ) ) ) | |
| 4 | 3 | al2imi | ⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) → ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜓 ) ) ) |
| 5 | sb2vOLD | ⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜓 ) → [ 𝑦 / 𝑥 ] 𝜓 ) | |
| 6 | 2 4 5 | syl56 | ⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) → ( [ 𝑦 / 𝑥 ] 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) ) |
| 7 | 1 6 | syl | ⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) → ( [ 𝑦 / 𝑥 ] 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) ) |