Theorem hbsbc1gd 1980

Description: Deduction version of hbsbc1g 1945.

Hypotheses

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

Assertion

Ref	Expression
hbsbc1gd	|- ((ph /\ A e. B) -> ([A / x]ps -> A.x[A / x]ps))

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

Proof of Theorem hbsbc1gd

Step	Hyp	Ref	Expression
1		ax-4 972	. . . . . . . . 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 517	. . . . . . . 8 |- (ph -> (A.x y e. A <-> y e. A))
4	3	abbidv 1575	. . . . . . 7 |- (ph -> {y | A.x y e. A} = {y | y e. A})
5		eleq1 1532	. . . . . . . . 9 |- (y = z -> (y e. A <-> z e. A))
6	5	albidv 1277	. . . . . . . 8 |- (y = z -> (A.x y e. A <-> A.x z e. A))
7	6	cbvabv 1906	. . . . . . 7 |- {y | A.x y e. A} = {z | A.x z e. A}
8		abid2 1578	. . . . . . 7 |- {y | y e. A} = A
9	4, 7, 8	3eqtr3g 1528	. . . . . 6 |- (ph -> {z | A.x z e. A} = A)
10	9	eleq1d 1538	. . . . 5 |- (ph -> ({z | A.x z e. A} e. V <-> A e. V))
11	10	biimpar 417	. . . 4 |- ((ph /\ A e. V) -> {z | A.x z e. A} e. V)
12		hba1 1002	. . . . . 6 |- (A.x z e. A -> A.xA.x z e. A)
13	12	hbab 1466	. . . . 5 |- (y e. {z | A.x z e. A} -> A.x y e. {z | A.x z e. A})
14	13	hbsbc1g 1945	. . . 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 10	. . 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 1285	. . . . 5 |- (ph -> A.y(y e. A -> A.x y e. A))
17		abidhb 1909	. . . . 5 |- (A.y(y e. A -> A.x y e. A) -> {z | A.x z e. A} = A)
18		dfsbcq 1940	. . . . 5 |- ({z | A.x z e. A} = A -> ([{z | A.x z e. A} / x]ps <-> [A / x]ps))
19	16, 17, 18	3syl 20	. . . 4 |- (ph -> ([{z | A.x z e. A} / x]ps <-> [A / x]ps))
20	19	adantr 389	. . 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 12	. . . . . 6 |- (ph -> (ph -> A.xph))
23		ax-17 970	. . . . . . . 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 1911	. . . . . 6 |- (ph -> (A e. V -> A.x A e. V))
26	22, 25	hband 1110	. . . . 5 |- (ph -> ((ph /\ A e. V) -> A.x(ph /\ A e. V)))
27	26	anabsi5 495	. . . 4 |- ((ph /\ A e. V) -> A.x(ph /\ A e. V))
28	27, 20	albid 1103	. . 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 541	. 2 |- ((ph /\ A e. V) -> ([A / x]ps -> A.x[A / x]ps))
30		elisset 1814	. 2 |- (A e. B -> A e. V)
31	29, 30	sylan2 451	1 |- ((ph /\ A e. B) -> ([A / x]ps -> A.x[A / x]ps))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 953   = wceq 955   e. wcel 957  [wsbc 1169  {cab 1462  Vcvv 1808

This theorem is referenced by:  hbcsb1gd 2024

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809  df-sbc 1939

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