Theorem equequ2 1686

Description: An equivalence law for equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2017.)

Assertion

Ref	Expression
equequ2	⊢ (x = y → (z = x ↔ z = y))

Proof of Theorem equequ2

Step	Hyp	Ref	Expression
1		equequ1 1684	. 2 ⊢ (x = y → (x = z ↔ y = z))
2		equcom 1680	. 2 ⊢ (x = z ↔ z = x)
3		equcom 1680	. 2 ⊢ (y = z ↔ z = y)
4	1, 2, 3	3bitr3g 278	1 ⊢ (x = y → (z = x ↔ z = y))

Syntax hints:   → wi 4   ↔ wb 176

This theorem was proved from 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 theorem depends on definitions:  df-bi 177  df-ex 1542

This theorem is referenced by:  ax10lem2  1937  dveeq2  1940  ax10lem4  1941  ax9  1949  ax11v2  1992  ax11vALT  2097  dveeq2-o  2184  dveeq2-o16  2185  ax10-16  2190  ax11eq  2193  ax11inda  2200  ax11v2-o  2201  eujust  2206  eujustALT  2207  euf  2210  mo  2226  2mo  2282  2eu6  2289  iotaval  4351  nndisjeq  4430  dfid3  4769