Theorem freq2 2919

Description: Equality theorem for the founded predicate.

Assertion

Ref	Expression
freq2	|- (A = B -> (R Fr A <-> R Fr B))

Proof of Theorem freq2

Step	Hyp	Ref	Expression
1		frss 2917	. . . 4 |- (A (_ B -> (R Fr B -> R Fr A))
2		frss 2917	. . . 4 |- (B (_ A -> (R Fr A -> R Fr B))
3	1, 2	anim12i 333	. . 3 |- ((A (_ B /\ B (_ A) -> ((R Fr B -> R Fr A) /\ (R Fr A -> R Fr B)))
4		eqss 2074	. . 3 |- (A = B <-> (A (_ B /\ B (_ A))
5		bi 514	. . 3 |- ((R Fr B <-> R Fr A) <-> ((R Fr B -> R Fr A) /\ (R Fr A -> R Fr B)))
6	3, 4, 5	3imtr4 219	. 2 |- (A = B -> (R Fr B <-> R Fr A))
7	6	bicomd 520	1 |- (A = B -> (R Fr A <-> R Fr B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 955   (_ wss 2044   Fr wfr 2911

This theorem is referenced by:  efrirr 2924  weeq2 2934  f1oweALT 3901

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-or 224  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-sn 2409  df-pr 2410  df-op 2413  df-br 2616  df-fr 2913

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