Theorem elequ1 1713

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

Assertion

Ref	Expression
elequ1	⊢ (x = y → (x ∈ z ↔ y ∈ z))

Proof of Theorem elequ1

Step	Hyp	Ref	Expression
1		ax-13 1712	. 2 ⊢ (x = y → (x ∈ z → y ∈ z))
2		ax-13 1712	. . 3 ⊢ (y = x → (y ∈ z → x ∈ z))
3	2	equcoms 1681	. 2 ⊢ (x = y → (y ∈ z → x ∈ z))
4	1, 3	impbid 183	1 ⊢ (x = y → (x ∈ z ↔ y ∈ z))

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-13 1712

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

This theorem is used by:  ax11wdemo  1723  cleljust  2014  cleljustALT  2015  dveel1  2019  ax15  2021  elsb1  2103  ax11el  2194  ncfinlower  4484  tfinnn  4535  tz6.12-2  5347