Theorem sbc2or 1929

Description: The disjunction of two equivalences for class substitution does not require a class existence hypothesis.

Assertion

Ref	Expression
sbc2or	|- (([A / x]ph <-> E.x(x = A /\ ph)) \/ ([A / x]ph <-> A.x(x = A -> ph)))

Distinct variable group:   x,A

Proof of Theorem sbc2or

Step	Hyp	Ref	Expression
1		sbc5g 1925	. . 3 |- (A e. V -> ([A / x]ph <-> E.x(x = A /\ ph)))
2	1	orcd 272	. 2 |- (A e. V -> (([A / x]ph <-> E.x(x = A /\ ph)) \/ ([A / x]ph <-> A.x(x = A -> ph))))
3		pm5.15 663	. . 3 |- (([A / x]ph <-> E.x(x = A /\ ph)) \/ ([A / x]ph <-> -. E.x(x = A /\ ph)))
4		pm5.1 673	. . . . . 6 |- ((-. E.x(x = A /\ ph) /\ A.x(x = A -> ph)) -> (-. E.x(x = A /\ ph) <-> A.x(x = A -> ph)))
5		visset 1788	. . . . . . . . . 10 |- x e. V
6		eleq1 1510	. . . . . . . . . 10 |- (x = A -> (x e. V <-> A e. V))
7	5, 6	mpbii 193	. . . . . . . . 9 |- (x = A -> A e. V)
8	7	adantr 389	. . . . . . . 8 |- ((x = A /\ ph) -> A e. V)
9	8	con3i 98	. . . . . . 7 |- (-. A e. V -> -. (x = A /\ ph))
10	9	nexdv 1308	. . . . . 6 |- (-. A e. V -> -. E.x(x = A /\ ph))
11	7	con3i 98	. . . . . . . 8 |- (-. A e. V -> -. x = A)
12	11	pm2.21d 78	. . . . . . 7 |- (-. A e. V -> (x = A -> ph))
13	12	19.21aiv 1268	. . . . . 6 |- (-. A e. V -> A.x(x = A -> ph))
14	4, 10, 13	sylanc 471	. . . . 5 |- (-. A e. V -> (-. E.x(x = A /\ ph) <-> A.x(x = A -> ph)))
15	14	bibi2d 616	. . . 4 |- (-. A e. V -> (([A / x]ph <-> -. E.x(x = A /\ ph)) <-> ([A / x]ph <-> A.x(x = A -> ph))))
16	15	orbi2d 612	. . 3 |- (-. A e. V -> ((([A / x]ph <-> E.x(x = A /\ ph)) \/ ([A / x]ph <-> -. E.x(x = A /\ ph))) <-> (([A / x]ph <-> E.x(x = A /\ ph)) \/ ([A / x]ph <-> A.x(x = A -> ph)))))
17	3, 16	mpbii 193	. 2 |- (-. A e. V -> (([A / x]ph <-> E.x(x = A /\ ph)) \/ ([A / x]ph <-> A.x(x = A -> ph))))
18	2, 17	pm2.61i 126	1 |- (([A / x]ph <-> E.x(x = A /\ ph)) \/ ([A / x]ph <-> A.x(x = A -> ph)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223  A.wal 950  E.wex 956   = wceq 1099   e. wcel 1105  [wsbc 1153  Vcvv 1786

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-v 1787  df-sbc 1913

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