Theorem sb4or 1754

Description: One direction of a simplified definition of substitution when variables are distinct. Similar to sb4 1753 but stronger in intuitionistic logic. (Contributed by Jim Kingdon, 2-Feb-2018.)

Assertion

Ref	Expression
sb4or	|- ( A. x x = y \/ A. x ( [ y / x ] ph -> A. x ( x = y -> ph ) ) )

Proof of Theorem sb4or

Step	Hyp	Ref	Expression
1		equs5or 1751	. 2 |- ( A. x x = y \/ ( E. x ( x = y /\ ph ) -> A. x ( x = y -> ph ) ) )
2		nfe1 1425	. . . . . 6 |- F/ x E. x ( x = y /\ ph )
3		nfa1 1474	. . . . . 6 |- F/ x A. x ( x = y -> ph )
4	2, 3	nfim 1504	. . . . 5 |- F/ x ( E. x ( x = y /\ ph ) -> A. x ( x = y -> ph ) )
5	4	nfri 1452	. . . 4 |- ( ( E. x ( x = y /\ ph ) -> A. x ( x = y -> ph ) ) -> A. x ( E. x ( x = y /\ ph ) -> A. x ( x = y -> ph ) ) )
6		sb1 1689	. . . . 5 |- ( [ y / x ] ph -> E. x ( x = y /\ ph ) )
7	6	imim1i 59	. . . 4 |- ( ( E. x ( x = y /\ ph ) -> A. x ( x = y -> ph ) ) -> ( [ y / x ] ph -> A. x ( x = y -> ph ) ) )
8	5, 7	alrimih 1398	. . 3 |- ( ( E. x ( x = y /\ ph ) -> A. x ( x = y -> ph ) ) -> A. x ( [ y / x ] ph -> A. x ( x = y -> ph ) ) )
9	8	orim2i 710	. 2 |- ( ( A. x x = y \/ ( E. x ( x = y /\ ph ) -> A. x ( x = y -> ph ) ) ) -> ( A. x x = y \/ A. x ( [ y / x ] ph -> A. x ( x = y -> ph ) ) ) )
10	1, 9	ax-mp 7	1 |- ( A. x x = y \/ A. x ( [ y / x ] ph -> A. x ( x = y -> ph ) ) )

Syntax hints:   -> wi 4   /\ wa 102   \/ wo 661   A.wal 1282   E.wex 1421   [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686

This theorem is referenced by:  sb4bor  1756  nfsb2or  1758