Theorem nfsb2or 1809

Description: Bound-variable hypothesis builder for substitution. Similar to hbsb2 1808 but in intuitionistic logic a disjunction is stronger than an implication. (Contributed by Jim Kingdon, 2-Feb-2018.)

Assertion

Ref	Expression
nfsb2or	|- ( A. x x = y \/ F/ x [ y / x ] ph )

Proof of Theorem nfsb2or

Step	Hyp	Ref	Expression
1		sb4or 1805	. 2 |- ( A. x x = y \/ A. x ( [ y / x ] ph -> A. x ( x = y -> ph ) ) )
2		sb2 1740	. . . . . . 7 |- ( A. x ( x = y -> ph ) -> [ y / x ] ph )
3	2	a5i 1522	. . . . . 6 |- ( A. x ( x = y -> ph ) -> A. x [ y / x ] ph )
4	3	imim2i 12	. . . . 5 |- ( ( [ y / x ] ph -> A. x ( x = y -> ph ) ) -> ( [ y / x ] ph -> A. x [ y / x ] ph ) )
5	4	alimi 1431	. . . 4 |- ( A. x ( [ y / x ] ph -> A. x ( x = y -> ph ) ) -> A. x ( [ y / x ] ph -> A. x [ y / x ] ph ) )
6		df-nf 1437	. . . 4 |- ( F/ x [ y / x ] ph <-> A. x ( [ y / x ] ph -> A. x [ y / x ] ph ) )
7	5, 6	sylibr 133	. . 3 |- ( A. x ( [ y / x ] ph -> A. x ( x = y -> ph ) ) -> F/ x [ y / x ] ph )
8	7	orim2i 750	. 2 |- ( ( A. x x = y \/ A. x ( [ y / x ] ph -> A. x ( x = y -> ph ) ) ) -> ( A. x x = y \/ F/ x [ y / x ] ph ) )
9	1, 8	ax-mp 5	1 |- ( A. x x = y \/ F/ x [ y / x ] ph )

Syntax hints:   -> wi 4   \/ wo 697   A.wal 1329   F/wnf 1436   [wsb 1735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736

This theorem is referenced by:  sbequi  1811