Theorem bnj51 12426

Description: First-order logic and set theory.

Hypothesis

Ref	Expression
bnj51.1	|- B = {x e. A | -. ph}

Assertion

Ref	Expression
bnj51	|- -. E.y e. B [y / x]ph

Distinct variable group:   x,A

Proof of Theorem bnj51

Step	Hyp	Ref	Expression
1		ralnex 2113	. . 3 |- (A.y e. B -. [y / x]ph <-> -. E.y e. B [y / x]ph)
2		sbn 1601	. . . . 5 |- ([y / x] -. ph <-> -. [y / x]ph)
3	2	bicomi 189	. . . 4 |- (-. [y / x]ph <-> [y / x] -. ph)
4	3	ralbii 2127	. . 3 |- (A.y e. B -. [y / x]ph <-> A.y e. B [y / x] -. ph)
5	1, 4	bitr3i 192	. 2 |- (-. E.y e. B [y / x]ph <-> A.y e. B [y / x] -. ph)
6		bnj51.1	. . . . . 6 |- B = {x e. A | -. ph}
7	6	eleq2i 1961	. . . . 5 |- (y e. B <-> y e. {x e. A | -. ph})
8		df-rab 2112	. . . . . 6 |- {x e. A | -. ph} = {x | (x e. A /\ -. ph)}
9	8	eleq2i 1961	. . . . 5 |- (y e. {x e. A | -. ph} <-> y e. {x | (x e. A /\ -. ph)})
10	7, 9	bitri 190	. . . 4 |- (y e. B <-> y e. {x | (x e. A /\ -. ph)})
11		df-clab 1872	. . . 4 |- (y e. {x | (x e. A /\ -. ph)} <-> [y / x](x e. A /\ -. ph))
12		bnj16 12380	. . . 4 |- ([y / x](x e. A /\ -. ph) <-> (y e. A /\ [y / x] -. ph))
13	10, 11, 12	3bitri 194	. . 3 |- (y e. B <-> (y e. A /\ [y / x] -. ph))
14	13	simprbi 353	. 2 |- (y e. B -> [y / x] -. ph)
15	5, 14	mprgbir 2163	1 |- -. E.y e. B [y / x]ph

Syntax hints:  -. wn 2   /\ wa 240   = wceq 1298   e. wcel 1300  [wsbc 1534  {cab 1871  A.wral 2105  E.wrex 2106  {crab 2108

This theorem is referenced by:  bnj12 13194  bnj56 13195

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-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-ral 2109  df-rex 2110  df-rab 2112

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