Theorem pm13.14 38610

Description: Theorem *13.14 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)

Assertion

Ref	Expression
pm13.14	|- ( ( [. A / x ]. ph /\ -. ph ) -> x =/= A )

Proof of Theorem pm13.14

Step	Hyp	Ref	Expression
1		sbceq1a 3446	. . . 4 |- ( x = A -> ( ph <-> [. A / x ]. ph ) )
2	1	biimprcd 240	. . 3 |- ( [. A / x ]. ph -> ( x = A -> ph ) )
3	2	necon3bd 2808	. 2 |- ( [. A / x ]. ph -> ( -. ph -> x =/= A ) )
4	3	imp 445	1 |- ( ( [. A / x ]. ph /\ -. ph ) -> x =/= A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   =/= wne 2794   [.wsbc 3435

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-9 1999  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ne 2795  df-sbc 3436

This theorem is referenced by: (None)