Theorem sbc2or 2481

Description: The disjunction of two equivalences for class substitution does not require a class existence hypothesis. This theorem tells us that there are only 2 possibilities for defining [A / x]ph behavior at proper classes (given a specific A), which is interesting since dfsbcq 2455 (from which it is derived) does not say anything obvious about that.

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 2470	. . 3 |- (A e. _V -> ([A / x]ph <-> E.x(x = A /\ ph)))
2	1	orcd 294	. 2 |- (A e. _V -> (([A / x]ph <-> E.x(x = A /\ ph)) \/ ([A / x]ph <-> A.x(x = A -> ph))))
3		pm5.15 729	. . 3 |- (([A / x]ph <-> E.x(x = A /\ ph)) \/ ([A / x]ph <-> -. E.x(x = A /\ ph)))
4		visset 2295	. . . . . . . . . 10 |- x e. _V
5		eleq1 1957	. . . . . . . . . 10 |- (x = A -> (x e. _V <-> A e. _V))
6	4, 5	mpbii 210	. . . . . . . . 9 |- (x = A -> A e. _V)
7	6	adantr 425	. . . . . . . 8 |- ((x = A /\ ph) -> A e. _V)
8	7	con3i 114	. . . . . . 7 |- (-. A e. _V -> -. (x = A /\ ph))
9	8	nexdv 1711	. . . . . 6 |- (-. A e. _V -> -. E.x(x = A /\ ph))
10	6	con3i 114	. . . . . . . 8 |- (-. A e. _V -> -. x = A)
11	10	pm2.21d 94	. . . . . . 7 |- (-. A e. _V -> (x = A -> ph))
12	11	19.21aiv 1664	. . . . . 6 |- (-. A e. _V -> A.x(x = A -> ph))
13		pm5.1 740	. . . . . 6 |- ((-. E.x(x = A /\ ph) /\ A.x(x = A -> ph)) -> (-. E.x(x = A /\ ph) <-> A.x(x = A -> ph)))
14	9, 12, 13	syl11anc 524	. . . . 5 |- (-. A e. _V -> (-. E.x(x = A /\ ph) <-> A.x(x = A -> ph)))
15	14	bibi2d 680	. . . 4 |- (-. A e. _V -> (([A / x]ph <-> -. E.x(x = A /\ ph)) <-> ([A / x]ph <-> A.x(x = A -> ph))))
16	15	orbi2d 676	. . 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 210	. 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 140	1 |- (([A / x]ph <-> E.x(x = A /\ ph)) \/ ([A / x]ph <-> A.x(x = A -> ph)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  [wsbc 1534  _Vcvv 2292

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-sbc 2454

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