Theorem hbsbc1gdOLD 2516

Description: Deduction version of hbsbc1g 2461.

Hypotheses

Ref	Expression
hbsbc1gd.1	|- (ph -> A.xph)
hbsbc1gd.2	|- (ph -> (y e. A -> A.x y e. A))

Assertion

Ref	Expression
hbsbc1gdOLD	|- ((ph /\ A e. B) -> ([A / x]ps -> A.x[A / x]ps))

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

Proof of Theorem hbsbc1gdOLD

Step	Hyp	Ref	Expression
1		ax-4 1319	. . . . . . . . 9 |- (A.x y e. A -> y e. A)
2		hbsbc1gd.2	. . . . . . . . 9 |- (ph -> (y e. A -> A.x y e. A))
3	1, 2	impbid2 576	. . . . . . . 8 |- (ph -> (A.x y e. A <-> y e. A))
4	3	abbidv 2008	. . . . . . 7 |- (ph -> {y | A.x y e. A} = {y | y e. A})
5		eleq1 1957	. . . . . . . . 9 |- (y = z -> (y e. A <-> z e. A))
6	5	albidv 1656	. . . . . . . 8 |- (y = z -> (A.x y e. A <-> A.x z e. A))
7	6	cbvabv 2420	. . . . . . 7 |- {y | A.x y e. A} = {z | A.x z e. A}
8		abid2 2011	. . . . . . 7 |- {y | y e. A} = A
9	4, 7, 8	3eqtr3g 1952	. . . . . 6 |- (ph -> {z | A.x z e. A} = A)
10	9	eleq1d 1963	. . . . 5 |- (ph -> ({z | A.x z e. A} e. _V <-> A e. _V))
11	10	biimpar 461	. . . 4 |- ((ph /\ A e. _V) -> {z | A.x z e. A} e. _V)
12		hba1 1350	. . . . . 6 |- (A.x z e. A -> A.xA.x z e. A)
13	12	hbab 1875	. . . . 5 |- (y e. {z | A.x z e. A} -> A.x y e. {z | A.x z e. A})
14	13	hbsbc1g 2461	. . . 4 |- ({z | A.x z e. A} e. _V -> ([{z | A.x z e. A} / x]ps -> A.x[{z | A.x z e. A} / x]ps))
15	11, 14	syl 12	. . 3 |- ((ph /\ A e. _V) -> ([{z | A.x z e. A} / x]ps -> A.x[{z | A.x z e. A} / x]ps))
16	2	19.21aiv 1664	. . . . 5 |- (ph -> A.y(y e. A -> A.x y e. A))
17		abidhb 2423	. . . . 5 |- (A.y(y e. A -> A.x y e. A) -> {z | A.x z e. A} = A)
18		dfsbcq 2455	. . . . 5 |- ({z | A.x z e. A} = A -> ([{z | A.x z e. A} / x]ps <-> [A / x]ps))
19	16, 17, 18	3syl 24	. . . 4 |- (ph -> ([{z | A.x z e. A} / x]ps <-> [A / x]ps))
20	19	adantr 425	. . 3 |- ((ph /\ A e. _V) -> ([{z | A.x z e. A} / x]ps <-> [A / x]ps))
21		hbsbc1gd.1	. . . . . . 7 |- (ph -> A.xph)
22	21	a1d 15	. . . . . 6 |- (ph -> (ph -> A.xph))
23		ax-17 1317	. . . . . . . 8 |- (y e. _V -> A.x y e. _V)
24	23	a1i 8	. . . . . . 7 |- (ph -> (y e. _V -> A.x y e. _V))
25	21, 2, 24	hbeld 2425	. . . . . 6 |- (ph -> (A e. _V -> A.x A e. _V))
26	22, 25	hband 1469	. . . . 5 |- (ph -> ((ph /\ A e. _V) -> A.x(ph /\ A e. _V)))
27	26	anabsi5 553	. . . 4 |- ((ph /\ A e. _V) -> A.x(ph /\ A e. _V))
28	27, 20	albid 1459	. . 3 |- ((ph /\ A e. _V) -> (A.x[{z | A.x z e. A} / x]ps <-> A.x[A / x]ps))
29	15, 20, 28	3imtr3d 601	. 2 |- ((ph /\ A e. _V) -> ([A / x]ps -> A.x[A / x]ps))
30		elisset 2299	. 2 |- (A e. B -> A e. _V)
31	29, 30	sylan2 500	1 |- ((ph /\ A e. B) -> ([A / x]ps -> A.x[A / x]ps))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  [wsbc 1534  {cab 1871  _Vcvv 2292

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  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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