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Mirrors > Home > MPE Home > Th. List > Mathboxes > mthmb | Structured version Visualization version GIF version |
Description: If two statements have the same reduct then one is a theorem iff the other is. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mthmb.r | ⊢ 𝑅 = (mStRed‘𝑇) |
mthmb.u | ⊢ 𝑈 = (mThm‘𝑇) |
Ref | Expression |
---|---|
mthmb | ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑈 ↔ 𝑌 ∈ 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mthmb.r | . . 3 ⊢ 𝑅 = (mStRed‘𝑇) | |
2 | mthmb.u | . . 3 ⊢ 𝑈 = (mThm‘𝑇) | |
3 | 1, 2 | mthmblem 31477 | . 2 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑈 → 𝑌 ∈ 𝑈)) |
4 | 1, 2 | mthmblem 31477 | . . 3 ⊢ ((𝑅‘𝑌) = (𝑅‘𝑋) → (𝑌 ∈ 𝑈 → 𝑋 ∈ 𝑈)) |
5 | 4 | eqcoms 2630 | . 2 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑌 ∈ 𝑈 → 𝑋 ∈ 𝑈)) |
6 | 3, 5 | impbid 202 | 1 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑈 ↔ 𝑌 ∈ 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 mStRedcmsr 31371 mThmcmthm 31376 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-ot 4186 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-1st 7168 df-2nd 7169 df-mpst 31390 df-msr 31391 df-mpps 31395 df-mthm 31396 |
This theorem is referenced by: (None) |
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