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Mirrors > Home > ILE Home > Th. List > impexp | GIF version |
Description: Import-export theorem. Part of Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 24-Mar-2013.) |
Ref | Expression |
---|---|
impexp | ⊢ (((𝜑 ∧ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 → 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm3.3 261 | . 2 ⊢ (((𝜑 ∧ 𝜓) → 𝜒) → (𝜑 → (𝜓 → 𝜒))) | |
2 | pm3.31 262 | . 2 ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 ∧ 𝜓) → 𝜒)) | |
3 | 1, 2 | impbii 126 | 1 ⊢ (((𝜑 ∧ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 → 𝜒))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: imp4a 349 exp4a 366 imdistan 444 pm5.3 475 pm4.87 557 nan 692 pm4.14dc 890 pm5.6dc 926 2sb6 1984 2sb6rf 1990 2exsb 2009 mor 2068 eu2 2070 moanim 2100 r2alf 2494 r3al 2521 r19.23t 2584 ceqsralt 2766 rspc2gv 2855 ralrab 2900 ralrab2 2904 euind 2926 reu2 2927 reu3 2929 rmo4 2932 rmo3f 2936 reuind 2944 rmo2ilem 3054 rmo3 3056 ralss 3223 rabss 3234 raldifb 3277 unissb 3841 elintrab 3858 ssintrab 3869 dftr5 4106 repizf2lem 4163 reusv3 4462 tfi 4583 raliunxp 4770 fununi 5286 dff13 5771 dfsmo2 6290 tfr1onlemaccex 6351 tfrcllemaccex 6364 qliftfun 6619 nnnninfeq2 7129 prime 9354 raluz 9580 raluz2 9581 ralrp 9677 facwordi 10722 modfsummod 11468 nnwosdc 12042 isprm2 12119 isprm4 12121 metcnp 14051 limcdifap 14170 bdcriota 14674 |
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