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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-evaleq | Structured version Visualization version GIF version |
Description: Equality theorem for the Slot construction. This is currently a duplicate of sloteq 17116 but may diverge from it if/when a token Eval is introduced for evaluation in order to separate it from Slot and any of its possible modifications. (Contributed by BJ, 27-Dec-2021.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-evaleq | ⊢ (𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6892 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑓‘𝐴) = (𝑓‘𝐵)) | |
2 | 1 | mpteq2dv 5251 | . 2 ⊢ (𝐴 = 𝐵 → (𝑓 ∈ V ↦ (𝑓‘𝐴)) = (𝑓 ∈ V ↦ (𝑓‘𝐵))) |
3 | df-slot 17115 | . 2 ⊢ Slot 𝐴 = (𝑓 ∈ V ↦ (𝑓‘𝐴)) | |
4 | df-slot 17115 | . 2 ⊢ Slot 𝐵 = (𝑓 ∈ V ↦ (𝑓‘𝐵)) | |
5 | 2, 3, 4 | 3eqtr4g 2798 | 1 ⊢ (𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 Vcvv 3475 ↦ cmpt 5232 ‘cfv 6544 Slot cslot 17114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-iota 6496 df-fv 6552 df-slot 17115 |
This theorem is referenced by: (None) |
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