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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > feq3dd | Structured version Visualization version GIF version |
Description: Equality deduction for functions. (Contributed by Thierry Arnoux, 27-May-2025.) |
Ref | Expression |
---|---|
feq3dd.eq | ⊢ (𝜑 → 𝐵 = 𝐶) |
feq3dd.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
Ref | Expression |
---|---|
feq3dd | ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | feq3dd.f | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
2 | feq3dd.eq | . . 3 ⊢ (𝜑 → 𝐵 = 𝐶) | |
3 | 2 | feq3d 6710 | . 2 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶𝐶)) |
4 | 1, 3 | mpbid 231 | 1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ⟶wf 6545 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1774 df-cleq 2717 df-ss 3961 df-f 6553 |
This theorem is referenced by: (None) |
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