Theorem sbeqal1 38598

Description: If x = y always implies x = z, then y = z is true. (Contributed by Andrew Salmon, 2-Jun-2011.)

Assertion

Ref	Expression
sbeqal1	|- ( A. x ( x = y -> x = z ) -> y = z )

Distinct variable group:   x, z

Proof of Theorem sbeqal1

Step	Hyp	Ref	Expression
1		sb2 2352	. 2 |- ( A. x ( x = y -> x = z ) -> [ y / x ] x = z )
2		equsb3 2432	. 2 |- ( [ y / x ] x = z <-> y = z )
3	1, 2	sylib 208	1 |- ( A. x ( x = y -> x = z ) -> y = z )

Syntax hints:   -> wi 4   A.wal 1481   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  sbeqal1i  38599  sbeqalbi  38601