Theorem r19.12sn 2444

Description: Special case of r19.12 1740 where its converse holds.

Hypothesis

Ref	Expression
r19.12sn.1	|- A e. V

Assertion

Ref	Expression
r19.12sn	|- (E.x e. {A}A.y e. B ph <-> A.y e. B E.x e. {A}ph)

Distinct variable groups:   x,y,A   x,B

Proof of Theorem r19.12sn

Step	Hyp	Ref	Expression
1		r19.12sn.1	. . . 4 |- A e. V
2	1, 1	rexpr 2429	. . 3 |- (E.x e. {A, A}A.y e. B ph <-> ([A / x]A.y e. B ph \/ [A / x]A.y e. B ph))
3		oridm 243	. . 3 |- (([A / x]A.y e. B ph \/ [A / x]A.y e. B ph) <-> [A / x]A.y e. B ph)
4		sbcralg 1994	. . . 4 |- (A e. V -> ([A / x]A.y e. B ph <-> A.y e. B [A / x]ph))
5	1, 4	ax-mp 7	. . 3 |- ([A / x]A.y e. B ph <-> A.y e. B [A / x]ph)
6	2, 3, 5	3bitr 177	. 2 |- (E.x e. {A, A}A.y e. B ph <-> A.y e. B [A / x]ph)
7		dfsn2 2420	. . 3 |- {A} = {A, A}
8		rexeq1 1787	. . 3 |- ({A} = {A, A} -> (E.x e. {A}A.y e. B ph <-> E.x e. {A, A}A.y e. B ph))
9	7, 8	ax-mp 7	. 2 |- (E.x e. {A}A.y e. B ph <-> E.x e. {A, A}A.y e. B ph)
10		rexeq1 1787	. . . . 5 |- ({A} = {A, A} -> (E.x e. {A}ph <-> E.x e. {A, A}ph))
11	7, 10	ax-mp 7	. . . 4 |- (E.x e. {A}ph <-> E.x e. {A, A}ph)
12	1, 1	rexpr 2429	. . . 4 |- (E.x e. {A, A}ph <-> ([A / x]ph \/ [A / x]ph))
13		oridm 243	. . . 4 |- (([A / x]ph \/ [A / x]ph) <-> [A / x]ph)
14	11, 12, 13	3bitr 177	. . 3 |- (E.x e. {A}ph <-> [A / x]ph)
15	14	ralbii 1667	. 2 |- (A.y e. B E.x e. {A}ph <-> A.y e. B [A / x]ph)
16	6, 9, 15	3bitr4 183	1 |- (E.x e. {A}A.y e. B ph <-> A.y e. B E.x e. {A}ph)

Syntax hints:   <-> wb 146   \/ wo 222   = wceq 956   e. wcel 958  [wsbc 1170  A.wral 1645  E.wrex 1646  Vcvv 1811  {csn 2409  {cpr 2410

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-rex 1650  df-v 1812  df-sbc 1942  df-un 2050  df-sn 2412  df-pr 2413

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