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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 6719 | . 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷)) | |
| 4 | 1, 2, 3 | mp2an 692 | 1 ⊢ (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ⟶wf 6557 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-ss 3968 df-fn 6564 df-f 6565 |
| This theorem is referenced by: ftpg 7176 hashf 14377 funcoppc 17920 cnextfval 24070 uhgr0 29090 lfgredgge2 29141 mbfmvolf 34268 eulerpartlemt 34373 ismgmOLD 37857 elghomOLD 37894 tendoset 40761 pwssplit4 43101 gricushgr 47886 uspgrlimlem2 47956 lincdifsn 48341 |
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