Axiom ax-14 974

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 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.

Assertion

Ref	Expression
ax-14	⊢ (x = y → (z ∈ x → z ∈ y))

Detailed syntax breakdown of Axiom ax-14

Step	Hyp	Ref	Expression
1		vx	. . . 4 set x
2	1	cv 959	. . 3 class x
3		vy	. . . 4 set y
4	3	cv 959	. . 3 class y
5	2, 4	wceq 960	. 2 wff x = y
6		vz	. . . . 5 set z
7	6	cv 959	. . . 4 class z
8	7, 2	wcel 962	. . 3 wff z ∈ x
9	7, 4	wcel 962	. . 3 wff z ∈ y
10	8, 9	wi 3	. 2 wff (z ∈ x → z ∈ y)
11	5, 10	wi 3	1 wff (x = y → (z ∈ x → z ∈ y))

This axiom is referenced by:  elequ2 1141  dtruALT 2764  fv3 3749  elirrv 4613

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