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Mirrors > Home > ILE Home > Th. List > ax-14 | GIF version |
Description: Axiom of right equality for the binary predicate ∈. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the right-hand side of the ∈ binary predicate. Axiom scheme C13' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Axiom B8 (p. 75) of system S2 of [Tarski] p. 77. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
ax-14 | ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vx | . . 3 setvar 𝑥 | |
2 | vy | . . 3 setvar 𝑦 | |
3 | 1, 2 | weq 1496 | . 2 wff 𝑥 = 𝑦 |
4 | vz | . . . 4 setvar 𝑧 | |
5 | 4, 1 | wel 2142 | . . 3 wff 𝑧 ∈ 𝑥 |
6 | 4, 2 | wel 2142 | . . 3 wff 𝑧 ∈ 𝑦 |
7 | 5, 6 | wi 4 | . 2 wff (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦) |
8 | 3, 7 | wi 4 | 1 wff (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)) |
Colors of variables: wff set class |
This axiom is referenced by: elequ2 2146 el 4164 fv3 5519 |
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