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Theorem mthmb 35586
Description: If two statements have the same reduct then one is a theorem iff the other is. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mthmb.r 𝑅 = (mStRed‘𝑇)
mthmb.u 𝑈 = (mThm‘𝑇)
Assertion
Ref Expression
mthmb ((𝑅𝑋) = (𝑅𝑌) → (𝑋𝑈𝑌𝑈))

Proof of Theorem mthmb
StepHypRef Expression
1 mthmb.r . . 3 𝑅 = (mStRed‘𝑇)
2 mthmb.u . . 3 𝑈 = (mThm‘𝑇)
31, 2mthmblem 35585 . 2 ((𝑅𝑋) = (𝑅𝑌) → (𝑋𝑈𝑌𝑈))
41, 2mthmblem 35585 . . 3 ((𝑅𝑌) = (𝑅𝑋) → (𝑌𝑈𝑋𝑈))
54eqcoms 2745 . 2 ((𝑅𝑋) = (𝑅𝑌) → (𝑌𝑈𝑋𝑈))
63, 5impbid 212 1 ((𝑅𝑋) = (𝑅𝑌) → (𝑋𝑈𝑌𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  cfv 6561  mStRedcmsr 35479  mThmcmthm 35484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-ot 4635  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-1st 8014  df-2nd 8015  df-mpst 35498  df-msr 35499  df-mpps 35503  df-mthm 35504
This theorem is referenced by: (None)
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