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Theorem mthmi 31803
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 6354 . . . 4 (𝑥 = 𝑋 → (𝑅𝑥) = (𝑅𝑋))
21eqeq1d 2763 . . 3 (𝑥 = 𝑋 → ((𝑅𝑥) = (𝑅𝑌) ↔ (𝑅𝑋) = (𝑅𝑌)))
32rspcev 3450 . 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 31802 . 2 (𝑌𝑈 ↔ ∃𝑥𝐽 (𝑅𝑥) = (𝑅𝑌))
83, 7sylibr 224 1 ((𝑋𝐽 ∧ (𝑅𝑋) = (𝑅𝑌)) → 𝑌𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2140  wrex 3052  cfv 6050  mStRedcmsr 31700  mPPStcmpps 31704  mThmcmthm 31705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-ot 4331  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-ov 6818  df-oprab 6819  df-1st 7335  df-2nd 7336  df-mpst 31719  df-msr 31720  df-mpps 31724  df-mthm 31725
This theorem is referenced by:  mppsthm  31805  mthmblem  31806  mthmpps  31808
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