Axiom ax-14 805

Description: Axiom of Equality. One of the 3 non-logical predicate axioms of our predicate calculus. 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	. . 3 set x
2		vy	. . 3 set y
3	1, 2	weq 797	. 2 wff x = y
4		vz	. . . 4 set z
5	4, 1	wel 803	. . 3 wff z ∈ x
6	4, 2	wel 803	. . 3 wff z ∈ y
7	5, 6	wi 2	. 2 wff (z ∈ x → z ∈ y)
8	3, 7	wi 2	1 wff (x = y → (z ∈ x → z ∈ y))

This axiom is referenced by:  a14b 820  fv3 2839  eirrv 3449

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