Theorem sb6a 2448

Description: Equivalence for substitution. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Wolf Lammen, 23-Sep-2018.)

Assertion

Ref	Expression
sb6a	|- ( [ y / x ] ph <-> A. x ( x = y -> [ x / y ] ph ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem sb6a

Step	Hyp	Ref	Expression
1		sbco 2412	. 2 |- ( [ y / x ] [ x / y ] ph <-> [ y / x ] ph )
2		sb6 2429	. 2 |- ( [ y / x ] [ x / y ] ph <-> A. x ( x = y -> [ x / y ] ph ) )
3	1, 2	bitr3i 266	1 |- ( [ y / x ] ph <-> A. x ( x = y -> [ x / y ] ph ) )

Syntax hints:   -> wi 4   <-> wb 196   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: (None)