Theorem pm13.194 38613

Description: Theorem *13.194 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)

Assertion

Ref	Expression
pm13.194	|- ( ( ph /\ x = y ) <-> ( [ y / x ] ph /\ ph /\ x = y ) )

Proof of Theorem pm13.194

Step	Hyp	Ref	Expression
1		pm13.13a 38608	. . . 4 |- ( ( ph /\ x = y ) -> [. y / x ]. ph )
2		sbsbc 3439	. . . 4 |- ( [ y / x ] ph <-> [. y / x ]. ph )
3	1, 2	sylibr 224	. . 3 |- ( ( ph /\ x = y ) -> [ y / x ] ph )
4		simpl 473	. . 3 |- ( ( ph /\ x = y ) -> ph )
5		simpr 477	. . 3 |- ( ( ph /\ x = y ) -> x = y )
6	3, 4, 5	3jca 1242	. 2 |- ( ( ph /\ x = y ) -> ( [ y / x ] ph /\ ph /\ x = y ) )
7		3simpc 1060	. 2 |- ( ( [ y / x ] ph /\ ph /\ x = y ) -> ( ph /\ x = y ) )
8	6, 7	impbii 199	1 |- ( ( ph /\ x = y ) <-> ( [ y / x ] ph /\ ph /\ x = y ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   [wsb 1880   [.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-3an 1039  df-ex 1705  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-sbc 3436

This theorem is referenced by: (None)