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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 16476 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 6663 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑓‘𝐴) = (𝑓‘𝐵)) | |
2 | 1 | mpteq2dv 5153 | . 2 ⊢ (𝐴 = 𝐵 → (𝑓 ∈ V ↦ (𝑓‘𝐴)) = (𝑓 ∈ V ↦ (𝑓‘𝐵))) |
3 | df-slot 16475 | . 2 ⊢ Slot 𝐴 = (𝑓 ∈ V ↦ (𝑓‘𝐴)) | |
4 | df-slot 16475 | . 2 ⊢ Slot 𝐵 = (𝑓 ∈ V ↦ (𝑓‘𝐵)) | |
5 | 2, 3, 4 | 3eqtr4g 2878 | 1 ⊢ (𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 Vcvv 3492 ↦ cmpt 5137 ‘cfv 6348 Slot cslot 16470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-iota 6307 df-fv 6356 df-slot 16475 |
This theorem is referenced by: (None) |
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