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Description: A variable not free in a proposition remains so after substitution in that proposition with a distinct variable (inference associated with nfsb4t ). Theorem nfsb replaces the distinctor antecedent with a disjoint variable condition. See nfsbv for a weaker version of nfsb not requiring ax-13 . (Contributed by NM, 14-May-1993) (Revised by Mario Carneiro, 4-Oct-2016) Usage of this theorem is discouraged because it depends on ax-13 . Use nfsbv instead. (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | nfsb4.1 | ⊢ Ⅎ 𝑧 𝜑 | |
| Assertion | nfsb4 | ⊢ ( ¬ ∀ 𝑧 𝑧 = 𝑦 → Ⅎ 𝑧 [ 𝑦 / 𝑥 ] 𝜑 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfsb4.1 | ⊢ Ⅎ 𝑧 𝜑 | |
| 2 | nfsb4t | ⊢ ( ∀ 𝑥 Ⅎ 𝑧 𝜑 → ( ¬ ∀ 𝑧 𝑧 = 𝑦 → Ⅎ 𝑧 [ 𝑦 / 𝑥 ] 𝜑 ) ) | |
| 3 | 2 1 | mpg | ⊢ ( ¬ ∀ 𝑧 𝑧 = 𝑦 → Ⅎ 𝑧 [ 𝑦 / 𝑥 ] 𝜑 ) |