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Mirrors > Home > ILE Home > Th. List > feq123d | GIF version |
Description: Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.) |
Ref | Expression |
---|---|
feq12d.1 | ⊢ (𝜑 → 𝐹 = 𝐺) |
feq12d.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
feq123d.3 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
feq123d | ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | feq12d.1 | . . 3 ⊢ (𝜑 → 𝐹 = 𝐺) | |
2 | feq12d.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
3 | 1, 2 | feq12d 5354 | . 2 ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶)) |
4 | feq123d.3 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
5 | feq3 5349 | . . 3 ⊢ (𝐶 = 𝐷 → (𝐺:𝐵⟶𝐶 ↔ 𝐺:𝐵⟶𝐷)) | |
6 | 4, 5 | syl 14 | . 2 ⊢ (𝜑 → (𝐺:𝐵⟶𝐶 ↔ 𝐺:𝐵⟶𝐷)) |
7 | 3, 6 | bitrd 188 | 1 ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 ⟶wf 5211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-sn 3598 df-pr 3599 df-op 3601 df-br 4003 df-opab 4064 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-fun 5217 df-fn 5218 df-f 5219 |
This theorem is referenced by: feq123 5356 feq23d 5360 |
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