Theorem bj-dfsb2 32825

Description: Alternate (dual) definition of substitution df-sb 1881 not using dummy variables. (Contributed by BJ, 19-Mar-2021.)

Assertion

Ref	Expression
bj-dfsb2	|- ( [ y / x ] ph <-> ( A. x ( x = y -> ph ) \/ ( x = y /\ ph ) ) )

Proof of Theorem bj-dfsb2

Step	Hyp	Ref	Expression
1		df-sb 1881	. 2 |- ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )
2		bj-sbsb 32824	. 2 |- ( ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) <-> ( A. x ( x = y -> ph ) \/ ( x = y /\ ph ) ) )
3	1, 2	bitri 264	1 |- ( [ y / x ] ph <-> ( A. x ( x = y -> ph ) \/ ( x = y /\ ph ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   A.wal 1481   E.wex 1704   [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: (None)