Theorem weeq1 2937

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 2922	. . 3 |- (R = S -> (R Fr A <-> S Fr A))
2		soeq1 2853	. . 3 |- (R = S -> (R Or A <-> S Or A))
3	1, 2	anbi12d 628	. 2 |- (R = S -> ((R Fr A /\ R Or A) <-> (S Fr A /\ S Or A)))
4		df-we 2934	. 2 |- (R We A <-> (R Fr A /\ R Or A))
5		df-we 2934	. 2 |- (S We A <-> (S Fr A /\ S Or A))
6	3, 4, 5	3bitr4g 555	1 |- (R = S -> (R We A <-> S We A))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   Or wor 2839   Fr wfr 2915   We wwe 2916

This theorem is referenced by:  weth 4787

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-17 971  ax-4 973  ax-5o 975  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-ex 981  df-cleq 1469  df-clel 1472  df-ral 1649  df-rex 1650  df-br 2620  df-po 2840  df-so 2850  df-fr 2917  df-we 2934

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