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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 6067 | . 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷)) | |
4 | 1, 2, 3 | mp2an 708 | 1 ⊢ (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1523 ⟶wf 5922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 df-clel 2647 df-in 3614 df-ss 3621 df-fn 5929 df-f 5930 |
This theorem is referenced by: ftpg 6463 hashf 13165 funcoppc 16582 cnextfval 21913 uhgr0 26013 lfgredgge2 26064 mbfmvolf 30456 eulerpartlemt 30561 ismgmOLD 33779 elghomOLD 33816 tendoset 36364 pwssplit4 37976 lincdifsn 42538 |
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