Theorem bj-equsexval 32638

Description: Special case of equsexv 2109 proved from Tarski, ax-10 2019 (modal5) and hba1 2151 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-equsexval.1	|- ( x = y -> ( ph <-> A. x ps ) )

Assertion

Ref	Expression
bj-equsexval	|- ( E. x ( x = y /\ ph ) <-> A. x ps )

Distinct variable group:   x, y

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

Proof of Theorem bj-equsexval

Step	Hyp	Ref	Expression
1		bj-equsexval.1	. . . 4 |- ( x = y -> ( ph <-> A. x ps ) )
2	1	pm5.32i 669	. . 3 |- ( ( x = y /\ ph ) <-> ( x = y /\ A. x ps ) )
3	2	exbii 1774	. 2 |- ( E. x ( x = y /\ ph ) <-> E. x ( x = y /\ A. x ps ) )
4		ax6ev 1890	. . 3 |- E. x x = y
5		bj-19.41al 32637	. . 3 |- ( E. x ( x = y /\ A. x ps ) <-> ( E. x x = y /\ A. x ps ) )
6	4, 5	mpbiran 953	. 2 |- ( E. x ( x = y /\ A. x ps ) <-> A. x ps )
7	3, 6	bitri 264	1 |- ( E. x ( x = y /\ ph ) <-> A. x ps )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704

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

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

This theorem is referenced by: (None)