Axiom ax-14 2139

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.)

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 setvar 𝑥
2		vy	. . 3 setvar 𝑦
3	1, 2	weq 1491	. 2 wff 𝑥 = 𝑦
4		vz	. . . 4 setvar 𝑧
5	4, 1	wel 2137	. . 3 wff 𝑧 ∈ 𝑥
6	4, 2	wel 2137	. . 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))

This axiom is referenced by:  elequ2  2141  el  4157  fv3  5509