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Mirrors > Home > MPE Home > Th. List > sbss | Structured version Visualization version GIF version |
Description: Set substitution into the first argument of a subset relation. (Contributed by Rodolfo Medina, 7-Jul-2010.) (Proof shortened by Mario Carneiro, 14-Nov-2016.) |
Ref | Expression |
---|---|
sbss | ⊢ ([𝑦 / 𝑥]𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3203 | . 2 ⊢ 𝑦 ∈ V | |
2 | sbequ 2376 | . 2 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝑥 ⊆ 𝐴 ↔ [𝑦 / 𝑥]𝑥 ⊆ 𝐴)) | |
3 | sseq1 3626 | . 2 ⊢ (𝑧 = 𝑦 → (𝑧 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴)) | |
4 | nfv 1843 | . . 3 ⊢ Ⅎ𝑥 𝑧 ⊆ 𝐴 | |
5 | sseq1 3626 | . . 3 ⊢ (𝑥 = 𝑧 → (𝑥 ⊆ 𝐴 ↔ 𝑧 ⊆ 𝐴)) | |
6 | 4, 5 | sbie 2408 | . 2 ⊢ ([𝑧 / 𝑥]𝑥 ⊆ 𝐴 ↔ 𝑧 ⊆ 𝐴) |
7 | 1, 2, 3, 6 | vtoclb 3263 | 1 ⊢ ([𝑦 / 𝑥]𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 [wsb 1880 ⊆ wss 3574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-v 3202 df-in 3581 df-ss 3588 |
This theorem is referenced by: (None) |
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