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Theorem mthmi 31235
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 6158 . . . 4 (𝑥 = 𝑋 → (𝑅𝑥) = (𝑅𝑋))
21eqeq1d 2623 . . 3 (𝑥 = 𝑋 → ((𝑅𝑥) = (𝑅𝑌) ↔ (𝑅𝑋) = (𝑅𝑌)))
32rspcev 3299 . 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 31234 . 2 (𝑌𝑈 ↔ ∃𝑥𝐽 (𝑅𝑥) = (𝑅𝑌))
83, 7sylibr 224 1 ((𝑋𝐽 ∧ (𝑅𝑋) = (𝑅𝑌)) → 𝑌𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wrex 2909  cfv 5857  mStRedcmsr 31132  mPPStcmpps 31136  mThmcmthm 31137
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-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-ot 4164  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-1st 7128  df-2nd 7129  df-mpst 31151  df-msr 31152  df-mpps 31156  df-mthm 31157
This theorem is referenced by:  mppsthm  31237  mthmblem  31238  mthmpps  31240
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