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Theorem trssOLD 4732
Description: Obsolete proof of trss 4731 as of 26-Jul-2021. (Contributed by NM, 7-Aug-1994.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
trssOLD (Tr 𝐴 → (𝐵𝐴𝐵𝐴))

Proof of Theorem trssOLD
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2686 . . . . 5 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
2 sseq1 3611 . . . . 5 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
31, 2imbi12d 334 . . . 4 (𝑥 = 𝐵 → ((𝑥𝐴𝑥𝐴) ↔ (𝐵𝐴𝐵𝐴)))
43imbi2d 330 . . 3 (𝑥 = 𝐵 → ((Tr 𝐴 → (𝑥𝐴𝑥𝐴)) ↔ (Tr 𝐴 → (𝐵𝐴𝐵𝐴))))
5 dftr3 4726 . . . 4 (Tr 𝐴 ↔ ∀𝑥𝐴 𝑥𝐴)
6 rsp 2925 . . . 4 (∀𝑥𝐴 𝑥𝐴 → (𝑥𝐴𝑥𝐴))
75, 6sylbi 207 . . 3 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
84, 7vtoclg 3256 . 2 (𝐵𝐴 → (Tr 𝐴 → (𝐵𝐴𝐵𝐴)))
98pm2.43b 55 1 (Tr 𝐴 → (𝐵𝐴𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  wral 2908  wss 3560  Tr wtr 4722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-v 3192  df-in 3567  df-ss 3574  df-uni 4410  df-tr 4723
This theorem is referenced by: (None)
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