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Theorem mthmi 32937
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 6654 . . 3 (𝑥 = 𝑋 → ((𝑅𝑥) = (𝑅𝑌) ↔ (𝑅𝑋) = (𝑅𝑌)))
21rspcev 3571 . 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 32936 . 2 (𝑌𝑈 ↔ ∃𝑥𝐽 (𝑅𝑥) = (𝑅𝑌))
72, 6sylibr 237 1 ((𝑋𝐽 ∧ (𝑅𝑋) = (𝑅𝑌)) → 𝑌𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2111  wrex 3107  cfv 6324  mStRedcmsr 32834  mPPStcmpps 32838  mThmcmthm 32839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-ot 4534  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-1st 7671  df-2nd 7672  df-mpst 32853  df-msr 32854  df-mpps 32858  df-mthm 32859
This theorem is referenced by:  mppsthm  32939  mthmblem  32940  mthmpps  32942
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