Description: Substitution analogue of Theorem 19.20 of Margaris p. 90 ( alim ). (Contributed by NM, 11-Nov-2005) (Revised by NM, 17-Aug-2018) (Proof shortened by JJ, 7-Jul-2021)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | sbcimdv.1 | ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) ) | |
| Assertion | sbcimdv | ⊢ ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 → [ 𝐴 / 𝑥 ] 𝜒 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sbcimdv.1 | ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) ) | |
| 2 | sbcex | ⊢ ( [ 𝐴 / 𝑥 ] 𝜓 → 𝐴 ∈ V ) | |
| 3 | 1 | alrimiv | ⊢ ( 𝜑 → ∀ 𝑥 ( 𝜓 → 𝜒 ) ) |
| 4 | spsbc | ⊢ ( 𝐴 ∈ V → ( ∀ 𝑥 ( 𝜓 → 𝜒 ) → [ 𝐴 / 𝑥 ] ( 𝜓 → 𝜒 ) ) ) | |
| 5 | sbcim1 | ⊢ ( [ 𝐴 / 𝑥 ] ( 𝜓 → 𝜒 ) → ( [ 𝐴 / 𝑥 ] 𝜓 → [ 𝐴 / 𝑥 ] 𝜒 ) ) | |
| 6 | 3 4 5 | syl56 | ⊢ ( 𝐴 ∈ V → ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 → [ 𝐴 / 𝑥 ] 𝜒 ) ) ) |
| 7 | 6 | com3l | ⊢ ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 → ( 𝐴 ∈ V → [ 𝐴 / 𝑥 ] 𝜒 ) ) ) |
| 8 | 2 7 | mpdi | ⊢ ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 → [ 𝐴 / 𝑥 ] 𝜒 ) ) |