Axiom ax-13 2138

Description: Axiom of left 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 left-hand side of the ∈ binary predicate. Axiom scheme C12' 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. "Non-logical" means that the predicate is not a primitive of predicate calculus proper but instead is an extension to it. "Binary" means that the predicate has two arguments. In a system of predicate calculus with equality, like ours, equality is not usually considered to be a non-logical predicate. In systems of predicate calculus without equality, it typically would be. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
ax-13	⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))

Detailed syntax breakdown of Axiom ax-13

Step	Hyp	Ref	Expression
1		vx	. . 3 setvar 𝑥
2		vy	. . 3 setvar 𝑦
3	1, 2	weq 1491	. 2 wff 𝑥 = 𝑦
4		vz	. . . 4 setvar 𝑧
5	1, 4	wel 2137	. . 3 wff 𝑥 ∈ 𝑧
6	2, 4	wel 2137	. . 3 wff 𝑦 ∈ 𝑧
7	5, 6	wi 4	. 2 wff (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)
8	3, 7	wi 4	1 wff (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))

This axiom is referenced by:  elequ1  2140  el  4156