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Mirrors > Home > ILE Home > Th. List > feq23d | GIF version |
Description: Equality deduction for functions. (Contributed by NM, 8-Jun-2013.) |
Ref | Expression |
---|---|
feq23d.1 | ⊢ (𝜑 → 𝐴 = 𝐶) |
feq23d.2 | ⊢ (𝜑 → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
feq23d | ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2165 | . 2 ⊢ (𝜑 → 𝐹 = 𝐹) | |
2 | feq23d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐶) | |
3 | feq23d.2 | . 2 ⊢ (𝜑 → 𝐵 = 𝐷) | |
4 | 1, 2, 3 | feq123d 5322 | 1 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1342 ⟶wf 5178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-fun 5184 df-fn 5185 df-f 5186 |
This theorem is referenced by: (None) |
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