Theorem bj-sbsb 32824

Description: Biconditional showing two possible (dual) definitions of substitution df-sb 1881 not using dummy variables. (Contributed by BJ, 19-Mar-2021.)

Assertion

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

Proof of Theorem bj-sbsb

Step	Hyp	Ref	Expression
1		simpl 473	. . . 4 |- ( ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) -> ( x = y -> ph ) )
2		pm2.27 42	. . . . . 6 |- ( x = y -> ( ( x = y -> ph ) -> ph ) )
3	2	anc2li 580	. . . . 5 |- ( x = y -> ( ( x = y -> ph ) -> ( x = y /\ ph ) ) )
4	3	sps 2055	. . . 4 |- ( A. x x = y -> ( ( x = y -> ph ) -> ( x = y /\ ph ) ) )
5		olc 399	. . . 4 |- ( ( x = y /\ ph ) -> ( A. x ( x = y -> ph ) \/ ( x = y /\ ph ) ) )
6	1, 4, 5	syl56 36	. . 3 |- ( A. x x = y -> ( ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) -> ( A. x ( x = y -> ph ) \/ ( x = y /\ ph ) ) ) )
7		simpr 477	. . . 4 |- ( ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) -> E. x ( x = y /\ ph ) )
8		equs5 2351	. . . . 5 |- ( -. A. x x = y -> ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) ) )
9	8	biimpd 219	. . . 4 |- ( -. A. x x = y -> ( E. x ( x = y /\ ph ) -> A. x ( x = y -> ph ) ) )
10		orc 400	. . . 4 |- ( A. x ( x = y -> ph ) -> ( A. x ( x = y -> ph ) \/ ( x = y /\ ph ) ) )
11	7, 9, 10	syl56 36	. . 3 |- ( -. A. x x = y -> ( ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) -> ( A. x ( x = y -> ph ) \/ ( x = y /\ ph ) ) ) )
12	6, 11	pm2.61i 176	. 2 |- ( ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) -> ( A. x ( x = y -> ph ) \/ ( x = y /\ ph ) ) )
13		sp 2053	. . . 4 |- ( A. x ( x = y -> ph ) -> ( x = y -> ph ) )
14		pm3.4 584	. . . 4 |- ( ( x = y /\ ph ) -> ( x = y -> ph ) )
15	13, 14	jaoi 394	. . 3 |- ( ( A. x ( x = y -> ph ) \/ ( x = y /\ ph ) ) -> ( x = y -> ph ) )
16		equs4 2290	. . . 4 |- ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) )
17		19.8a 2052	. . . 4 |- ( ( x = y /\ ph ) -> E. x ( x = y /\ ph ) )
18	16, 17	jaoi 394	. . 3 |- ( ( A. x ( x = y -> ph ) \/ ( x = y /\ ph ) ) -> E. x ( x = y /\ ph ) )
19	15, 18	jca 554	. 2 |- ( ( A. x ( x = y -> ph ) \/ ( x = y /\ ph ) ) -> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )
20	12, 19	impbii 199	1 |- ( ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) <-> ( A. x ( x = y -> ph ) \/ ( x = y /\ ph ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ 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  ax-13 2246

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:  bj-dfsb2  32825