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Theorem mthmi 30728
 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 fveq2 6103 . . . 4 (𝑥 = 𝑋 → (𝑅𝑥) = (𝑅𝑋))
21eqeq1d 2612 . . 3 (𝑥 = 𝑋 → ((𝑅𝑥) = (𝑅𝑌) ↔ (𝑅𝑋) = (𝑅𝑌)))
32rspcev 3282 . 2 ((𝑋𝐽 ∧ (𝑅𝑋) = (𝑅𝑌)) → ∃𝑥𝐽 (𝑅𝑥) = (𝑅𝑌))
4 mthmval.r . . 3 𝑅 = (mStRed‘𝑇)
5 mthmval.j . . 3 𝐽 = (mPPSt‘𝑇)
6 mthmval.u . . 3 𝑈 = (mThm‘𝑇)
74, 5, 6elmthm 30727 . 2 (𝑌𝑈 ↔ ∃𝑥𝐽 (𝑅𝑥) = (𝑅𝑌))
83, 7sylibr 223 1 ((𝑋𝐽 ∧ (𝑅𝑋) = (𝑅𝑌)) → 𝑌𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  ‘cfv 5804  mStRedcmsr 30625  mPPStcmpps 30629  mThmcmthm 30630 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-ot 4134  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-1st 7059  df-2nd 7060  df-mpst 30644  df-msr 30645  df-mpps 30649  df-mthm 30650 This theorem is referenced by:  mppsthm  30730  mthmblem  30731  mthmpps  30733
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