Theorem elequ2 1715

Description: An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
elequ2	⊢ (x = y → (z ∈ x ↔ z ∈ y))

Proof of Theorem elequ2

Step	Hyp	Ref	Expression
1		ax-14 1714	. 2 ⊢ (x = y → (z ∈ x → z ∈ y))
2		ax-14 1714	. . 3 ⊢ (y = x → (z ∈ y → z ∈ x))
3	2	equcoms 1681	. 2 ⊢ (x = y → (z ∈ y → z ∈ x))
4	1, 3	impbid 183	1 ⊢ (x = y → (z ∈ x ↔ z ∈ y))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714

This proof depends on definitions:  df-bi 177  df-ex 1542

This theorem is used by:  ax11wdemo  1723  dveel2  2020  elsb2  2104  dveel2ALT  2191  ax11el  2194  axext3  2336  axext4  2337  bm1.1  2338  ssfin  4471  ncfinlower  4484  nnadjoinlem1  4520  fv3  5342