Theorem trcleq3d 13931

Description: Equality deduction for the transitive closure.

Hypothesis

Ref	Expression
trcleq3d.1	|- (ph -> X = Y)

Assertion

Ref	Expression
trcleq3d	|- (ph -> Trcl(R, A, X) = Trcl(R, A, Y))

Proof of Theorem trcleq3d

Step	Hyp	Ref	Expression
1		trcleq3d.1	. 2 |- (ph -> X = Y)
2		trcleq3 13928	. 2 |- (X = Y -> Trcl(R, A, X) = Trcl(R, A, Y))
3	1, 2	syl 12	1 |- (ph -> Trcl(R, A, X) = Trcl(R, A, Y))

Syntax hints:   -> wi 3   = wceq 1298  Trclctrcl 13924

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-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  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  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-rdg 5140  df-pred 13880  df-trcl 13925

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