Theorem equequ1 1684

Description: An equivalence law for equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Dec-2017.)

Assertion

Ref	Expression
equequ1	⊢ (x = y → (x = z ↔ y = z))

Proof of Theorem equequ1

Step	Hyp	Ref	Expression
1		ax-8 1675	. 2 ⊢ (x = y → (x = z → y = z))
2		equtr 1682	. 2 ⊢ (x = y → (y = z → x = z))
3	1, 2	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

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

This theorem is used by:  equequ2  1686  ax12olem6  1932  ax10lem2  1937  ax10lem4  1941  equveli  1988  dveeq1  2018  drsb1  2022  equsb3lem  2101  dveeq1-o  2187  dveeq1-o16  2188  ax10-16  2190  ax11eq  2193  2mo  2282  2eu6  2289  euequ1  2292  axext3  2336  cbviota  4345