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Mirrors > Home > ILE Home > Th. List > feq2d | GIF version |
Description: Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.) |
Ref | Expression |
---|---|
feq2d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
feq2d | ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐹:𝐵⟶𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | feq2d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | feq2 5212 | . 2 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴⟶𝐶 ↔ 𝐹:𝐵⟶𝐶)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐹:𝐵⟶𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1312 ⟶wf 5075 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1404 ax-gen 1406 ax-4 1468 ax-17 1487 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-cleq 2106 df-fn 5082 df-f 5083 |
This theorem is referenced by: feq12d 5218 ffdm 5249 fsng 5545 issmo2 6138 qliftf 6466 elpm2r 6512 casef 6923 fseq1p1m1 9761 fseq1m1p1 9762 seqf 10121 seqf2 10124 lmtopcnp 12255 ellimc3apf 12579 dvidlemap 12609 dviaddf 12616 |
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