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Theorem hbcsb1gd 2570
Description: Deduction version of hbcsb1g 2567.
Hypotheses
Ref Expression
hbcsb1gd.1 |- (ph -> A.xph)
hbcsb1gd.2 |- (ph -> (y e. A -> A.x y e. A))
Assertion
Ref Expression
hbcsb1gd |- ((ph /\ A e. C) -> (y e. [_A / x]_B -> A.x y e. [_A / x]_B))
Distinct variable groups:   y,A   ph,y   x,y

Proof of Theorem hbcsb1gd
StepHypRef Expression
1 hbcsb1gd.1 . . . . . 6 |- (ph -> A.xph)
21a1d 15 . . . . 5 |- (ph -> (ph -> A.xph))
3 hbcsb1gd.2 . . . . . 6 |- (ph -> (y e. A -> A.x y e. A))
4 ax-17 1317 . . . . . . 7 |- (y e. _V -> A.x y e. _V)
54a1i 8 . . . . . 6 |- (ph -> (y e. _V -> A.x y e. _V))
61, 3, 5hbeld 2425 . . . . 5 |- (ph -> (A e. _V -> A.x A e. _V))
72, 6hband 1469 . . . 4 |- (ph -> ((ph /\ A e. _V) -> A.x(ph /\ A e. _V)))
87anabsi5 553 . . 3 |- ((ph /\ A e. _V) -> A.x(ph /\ A e. _V))
9 ax-17 1317 . . . 4 |- (z e. y -> A.x z e. y)
109a1i 8 . . 3 |- ((ph /\ A e. _V) -> (z e. y -> A.x z e. y))
111, 3hbsbc1gd 2515 . . . 4 |- ((ph /\ A e. _V) -> ([A / x]z e. B -> A.x[A / x]z e. B))
12 sbcel2g 2558 . . . . 5 |- (A e. _V -> ([A / x]z e. B <-> z e. [_A / x]_B))
1312adantl 424 . . . 4 |- ((ph /\ A e. _V) -> ([A / x]z e. B <-> z e. [_A / x]_B))
148, 13albid 1459 . . . 4 |- ((ph /\ A e. _V) -> (A.x[A / x]z e. B <-> A.x z e. [_A / x]_B))
1511, 13, 143imtr3d 601 . . 3 |- ((ph /\ A e. _V) -> (z e. [_A / x]_B -> A.x z e. [_A / x]_B))
168, 10, 15hbeld 2425 . 2 |- ((ph /\ A e. _V) -> (y e. [_A / x]_B -> A.x y e. [_A / x]_B))
17 elisset 2299 . 2 |- (A e. C -> A e. _V)
1816, 17sylan2 500 1 |- ((ph /\ A e. C) -> (y e. [_A / x]_B -> A.x y e. [_A / x]_B))
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   e. wcel 1300  [wsbc 1534  _Vcvv 2292  [_csb 2540
This theorem is referenced by:  csbnest1g 2582
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  df-csb 2541
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