Theorem sb4bor 1713

Description: Simplified definition of substitution when variables are distinct, expressed via disjunction. (Contributed by Jim Kingdon, 18-Mar-2018.)

Assertion

Ref	Expression
sb4bor	|- (A.x x = y \/ A.x([y / x]ph <-> A.x(x = y -> ph)))

Proof of Theorem sb4bor

Step	Hyp	Ref	Expression
1		sb4or 1711	. 2 |- (A.x x = y \/ A.x([y / x]ph -> A.x(x = y -> ph)))
2		sb2 1647	. . . . 5 |- (A.x(x = y -> ph) -> [y / x]ph)
3		df-bi 110	. . . . . 6 |- ((([y / x]ph <-> A.x(x = y -> ph)) -> (([y / x]ph -> A.x(x = y -> ph)) /\ (A.x(x = y -> ph) -> [y / x]ph))) /\ ((([y / x]ph -> A.x(x = y -> ph)) /\ (A.x(x = y -> ph) -> [y / x]ph)) -> ([y / x]ph <-> A.x(x = y -> ph))))
4	3	simpri 106	. . . . 5 |- ((([y / x]ph -> A.x(x = y -> ph)) /\ (A.x(x = y -> ph) -> [y / x]ph)) -> ([y / x]ph <-> A.x(x = y -> ph)))
5	2, 4	mpan2 401	. . . 4 |- (([y / x]ph -> A.x(x = y -> ph)) -> ([y / x]ph <-> A.x(x = y -> ph)))
6	5	alimi 1341	. . 3 |- (A.x([y / x]ph -> A.x(x = y -> ph)) -> A.x([y / x]ph <-> A.x(x = y -> ph)))
7	6	orim2i 677	. 2 |- ((A.x x = y \/ A.x([y / x]ph -> A.x(x = y -> ph))) -> (A.x x = y \/ A.x([y / x]ph <-> A.x(x = y -> ph))))
8	1, 7	ax-mp 7	1 |- (A.x x = y \/ A.x([y / x]ph <-> A.x(x = y -> ph)))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 628  A.wal 1240  [wsb 1642

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643

This theorem is referenced by: (None)