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Mirrors > Home > MPE Home > Th. List > feq23i | Structured version Visualization version GIF version |
Description: Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.) |
Ref | Expression |
---|---|
feq23i.1 | ⊢ 𝐴 = 𝐶 |
feq23i.2 | ⊢ 𝐵 = 𝐷 |
Ref | Expression |
---|---|
feq23i | ⊢ (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | feq23i.1 | . 2 ⊢ 𝐴 = 𝐶 | |
2 | feq23i.2 | . 2 ⊢ 𝐵 = 𝐷 | |
3 | feq23 6471 | . 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷)) | |
4 | 1, 2, 3 | mp2an 691 | 1 ⊢ (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1538 ⟶wf 6320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-in 3888 df-ss 3898 df-fn 6327 df-f 6328 |
This theorem is referenced by: ftpg 6895 hashf 13694 funcoppc 17137 cnextfval 22667 uhgr0 26866 lfgredgge2 26917 mbfmvolf 31634 eulerpartlemt 31739 ismgmOLD 35288 elghomOLD 35325 tendoset 38055 pwssplit4 40033 isomushgr 44344 lincdifsn 44833 |
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