Theorem weeq1 2927

Description: Equality theorem for the well-ordering predicate.

Assertion

Ref	Expression
weeq1	|- (R = S -> (R We A <-> S We A))

Proof of Theorem weeq1

Step	Hyp	Ref	Expression
1		freq1 2912	. . 3 |- (R = S -> (R Fr A <-> S Fr A))
2		soeq1 2844	. . 3 |- (R = S -> (R Or A <-> S Or A))
3	1, 2	anbi12d 626	. 2 |- (R = S -> ((R Fr A /\ R Or A) <-> (S Fr A /\ S Or A)))
4		df-we 2924	. 2 |- (R We A <-> (R Fr A /\ R Or A))
5		df-we 2924	. 2 |- (S We A <-> (S Fr A /\ S Or A))
6	3, 4, 5	3bitr4g 553	1 |- (R = S -> (R We A <-> S We A))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   Or wor 2830   Fr wfr 2905   We wwe 2906

This theorem is referenced by:  weth 4759

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-17 968  ax-4 970  ax-5o 972  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-ex 978  df-cleq 1462  df-clel 1465  df-ral 1641  df-rex 1642  df-br 2610  df-po 2831  df-so 2841  df-fr 2907  df-we 2924

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