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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 5942 | . 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷)) | |
4 | 1, 2, 3 | mp2an 704 | 1 ⊢ (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ⟶wf 5800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-in 3547 df-ss 3554 df-fn 5807 df-f 5808 |
This theorem is referenced by: ftpg 6328 hashf 12987 funcoppc 16358 cnextfval 21676 uhgr0 25739 lfgredgge2 25790 uhgra0v 25839 wlkntrllem1 26089 mbfmvolf 29655 eulerpartlemt 29760 ismgmOLD 32819 elghomOLD 32856 tendoset 35065 pwssplit4 36677 lincdifsn 42007 |
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