Theorem freq2 2923

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 2921	. . . 4 |- (A (_ B -> (R Fr B -> R Fr A))
2		frss 2921	. . . 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 2077	. . 3 |- (A = B <-> (A (_ B /\ B (_ A))
5		dfbi2 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 521	1 |- (A = B -> (R Fr A <-> R Fr B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   (_ wss 2047   Fr wfr 2915

This theorem is referenced by:  efrirr 2928  weeq2 2938  f1oweALT 3906

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-fr 2917

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