Theorem sbco2 1958

Description: A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)

Hypothesis

Ref	Expression
sbco2.1	|- F/ z ph

Assertion

Ref	Expression
sbco2	|- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )

Proof of Theorem sbco2

Step	Hyp	Ref	Expression
1		sbco2.1	. . 3 |- F/ z ph
2	1	nfri 1512	. 2 |- ( ph -> A. z ph )
3	2	sbco2h 1957	1 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )

Syntax hints:   <-> wb 104   F/wnf 1453   [wsb 1755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756

This theorem is referenced by:  nfsbt  1969  sb7af  1986  sbco4lem  1999  sbco4  2000  eqsb1  2274  clelsb1  2275  clelsb2  2276  sb8ab  2292  clelsb1f  2316  sbralie  2714  sbcco  2976