Theorem bnj42 13192

Description: Technical lemma of bnj7 13196. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem).

Hypotheses

Ref	Expression
bnj42.1	|- ((R Fr A /\ -. th) -> E.y e. B [y / x]ps)
bnj42.2	|- B C_ A

Assertion

Ref	Expression
bnj42	|- ((R Fr A /\ -. th) -> E.y e. A [y / x]ps)

Distinct variable groups:   y,A   y,B

Proof of Theorem bnj42

Step	Hyp	Ref	Expression
1		bnj42.1	. 2 |- ((R Fr A /\ -. th) -> E.y e. B [y / x]ps)
2		bnj42.2	. . 3 |- B C_ A
3	2	bnj46 12416	. 2 |- (E.y e. B [y / x]ps -> E.y e. A [y / x]ps)
4	1, 3	syl 12	1 |- ((R Fr A /\ -. th) -> E.y e. A [y / x]ps)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240  [wsbc 1534  E.wrex 2106   C_ wss 2593   Fr wfr 3623

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-10 1308  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-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-rex 2110  df-in 2603  df-ss 2605

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