Theorem mthmb 33571

Description: If two statements have the same reduct then one is a theorem iff the other is. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mthmb.r	⊢ 𝑅 = (mStRed‘𝑇)
mthmb.u	⊢ 𝑈 = (mThm‘𝑇)

Assertion

Ref	Expression
mthmb	⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑈 ↔ 𝑌 ∈ 𝑈))

Proof of Theorem mthmb

Step	Hyp	Ref	Expression
1		mthmb.r	. . 3 ⊢ 𝑅 = (mStRed‘𝑇)
2		mthmb.u	. . 3 ⊢ 𝑈 = (mThm‘𝑇)
3	1, 2	mthmblem 33570	. 2 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑈 → 𝑌 ∈ 𝑈))
4	1, 2	mthmblem 33570	. . 3 ⊢ ((𝑅‘𝑌) = (𝑅‘𝑋) → (𝑌 ∈ 𝑈 → 𝑋 ∈ 𝑈))
5	4	eqcoms 2741	. 2 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑌 ∈ 𝑈 → 𝑋 ∈ 𝑈))
6	3, 5	impbid 211	1 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑈 ↔ 𝑌 ∈ 𝑈))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1537   ∈ wcel 2101  ‘cfv 6447  mStRedcmsr 33464  mThmcmthm 33469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-rep 5212  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7608

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3223  df-rab 3224  df-v 3436  df-sbc 3719  df-csb 3835  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-pw 4538  df-sn 4565  df-pr 4567  df-op 4571  df-ot 4573  df-uni 4842  df-iun 4929  df-br 5078  df-opab 5140  df-mpt 5161  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-iota 6399  df-fun 6449  df-fn 6450  df-f 6451  df-f1 6452  df-fo 6453  df-f1o 6454  df-fv 6455  df-ov 7298  df-oprab 7299  df-1st 7851  df-2nd 7852  df-mpst 33483  df-msr 33484  df-mpps 33488  df-mthm 33489

This theorem is referenced by: (None)