Theorem wl-sbal2 33347

Description: Move quantifier in and out of substitution. Revised to remove a distinct variable constraint. (Contributed by NM, 2-Jan-2002.) Proof is based on wl-sbalnae 33345 now. See also sbal2 2461. (Revised by Wolf Lammen, 25-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
wl-sbal2	|- ( -. A. x x = y -> ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) )

Distinct variable group:   x, z

Allowed substitution hints:   ph( x, y, z)

Proof of Theorem wl-sbal2

Step	Hyp	Ref	Expression
1		wl-naev 33302	. 2 |- ( -. A. x x = y -> -. A. x x = z )
2		wl-sbalnae 33345	. 2 |- ( ( -. A. x x = y /\ -. A. x x = z ) -> ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) )
3	1, 2	mpdan 702	1 |- ( -. A. x x = y -> ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) )

Syntax hints:   -. wn 3   -> 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-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by: (None)