Axiom ax-13 968

Description: Axiom of Equality. 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.

Assertion

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

Detailed syntax breakdown of Axiom ax-13

Step	Hyp	Ref	Expression
1		vx	. . . 4 set x
2	1	cv 954	. . 3 class x
3		vy	. . . 4 set y
4	3	cv 954	. . 3 class y
5	2, 4	wceq 955	. 2 wff x = y
6		vz	. . . . 5 set z
7	6	cv 954	. . . 4 class z
8	2, 7	wcel 957	. . 3 wff x ∈ z
9	4, 7	wcel 957	. . 3 wff y ∈ z
10	8, 9	wi 3	. 2 wff (x ∈ z → y ∈ z)
11	5, 10	wi 3	1 wff (x = y → (x ∈ z → y ∈ z))

This axiom is referenced by:  elequ1 1135

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