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Mirrors > Home > ILE Home > Th. List > imim1i | GIF version |
Description: Inference adding common consequents in an implication, thereby interchanging the original antecedent and consequent. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2012.) |
Ref | Expression |
---|---|
imim1i.1 | ⊢ (𝜑 → 𝜓) |
Ref | Expression |
---|---|
imim1i | ⊢ ((𝜓 → 𝜒) → (𝜑 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imim1i.1 | . 2 ⊢ (𝜑 → 𝜓) | |
2 | id 19 | . 2 ⊢ (𝜒 → 𝜒) | |
3 | 1, 2 | imim12i 59 | 1 ⊢ ((𝜓 → 𝜒) → (𝜑 → 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 |
This theorem is referenced by: jarr 97 bi3ant 224 pm3.41 331 pm3.42 332 jarl 659 pm2.67-2 714 oibabs 715 stdcn 848 pm2.85dc 906 peircedc 915 3jaob 1313 hbim 1556 hbimd 1584 i19.39 1651 hbae 1729 sbcof2 1821 sb4or 1844 tfi 4599 dmcosseq 4916 fliftfun 5818 tfrcl 6389 ac6sfi 6926 fsum2d 11475 fsumabs 11505 fsumiun 11517 fprod2d 11663 bj-nnsn 14943 bj-pm2.18st 14960 setindis 15177 bdsetindis 15179 |
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