Description: Biconditional property for substitution. Closed form of sbbii . Specialization of biconditional. (Contributed by NM, 2-Jun-1993) Revise df-sb . (Revised by BJ, 22-Dec-2020)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | spsbbi | ⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) → ( [ 𝑡 / 𝑥 ] 𝜑 ↔ [ 𝑡 / 𝑥 ] 𝜓 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | biimp | ⊢ ( ( 𝜑 ↔ 𝜓 ) → ( 𝜑 → 𝜓 ) ) | |
| 2 | 1 | alimi | ⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) → ∀ 𝑥 ( 𝜑 → 𝜓 ) ) |
| 3 | spsbim | ⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( [ 𝑡 / 𝑥 ] 𝜑 → [ 𝑡 / 𝑥 ] 𝜓 ) ) | |
| 4 | 2 3 | syl | ⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) → ( [ 𝑡 / 𝑥 ] 𝜑 → [ 𝑡 / 𝑥 ] 𝜓 ) ) |
| 5 | biimpr | ⊢ ( ( 𝜑 ↔ 𝜓 ) → ( 𝜓 → 𝜑 ) ) | |
| 6 | 5 | alimi | ⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) → ∀ 𝑥 ( 𝜓 → 𝜑 ) ) |
| 7 | spsbim | ⊢ ( ∀ 𝑥 ( 𝜓 → 𝜑 ) → ( [ 𝑡 / 𝑥 ] 𝜓 → [ 𝑡 / 𝑥 ] 𝜑 ) ) | |
| 8 | 6 7 | syl | ⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) → ( [ 𝑡 / 𝑥 ] 𝜓 → [ 𝑡 / 𝑥 ] 𝜑 ) ) |
| 9 | 4 8 | impbid | ⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) → ( [ 𝑡 / 𝑥 ] 𝜑 ↔ [ 𝑡 / 𝑥 ] 𝜓 ) ) |