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Theorem mthmi 35561
Description: A statement whose reduct is the reduct of a provable pre-statement is a theorem. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mthmval.r 𝑅 = (mStRed‘𝑇)
mthmval.j 𝐽 = (mPPSt‘𝑇)
mthmval.u 𝑈 = (mThm‘𝑇)
Assertion
Ref Expression
mthmi ((𝑋𝐽 ∧ (𝑅𝑋) = (𝑅𝑌)) → 𝑌𝑈)

Proof of Theorem mthmi
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fveqeq2 6915 . . 3 (𝑥 = 𝑋 → ((𝑅𝑥) = (𝑅𝑌) ↔ (𝑅𝑋) = (𝑅𝑌)))
21rspcev 3621 . 2 ((𝑋𝐽 ∧ (𝑅𝑋) = (𝑅𝑌)) → ∃𝑥𝐽 (𝑅𝑥) = (𝑅𝑌))
3 mthmval.r . . 3 𝑅 = (mStRed‘𝑇)
4 mthmval.j . . 3 𝐽 = (mPPSt‘𝑇)
5 mthmval.u . . 3 𝑈 = (mThm‘𝑇)
63, 4, 5elmthm 35560 . 2 (𝑌𝑈 ↔ ∃𝑥𝐽 (𝑅𝑥) = (𝑅𝑌))
72, 6sylibr 234 1 ((𝑋𝐽 ∧ (𝑅𝑋) = (𝑅𝑌)) → 𝑌𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  wrex 3067  cfv 6562  mStRedcmsr 35458  mPPStcmpps 35462  mThmcmthm 35463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-ot 4639  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-1st 8012  df-2nd 8013  df-mpst 35477  df-msr 35478  df-mpps 35482  df-mthm 35483
This theorem is referenced by:  mppsthm  35563  mthmblem  35564  mthmpps  35566
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