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Mirrors > Home > MPE Home > Th. List > elsb4 | Structured version Visualization version GIF version |
Description: Substitution applied to an atomic membership wff. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
Ref | Expression |
---|---|
elsb4 | ⊢ ([𝑥 / 𝑦]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1830 | . . 3 ⊢ Ⅎ𝑦 𝑧 ∈ 𝑤 | |
2 | 1 | sbco2 2403 | . 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑤]𝑧 ∈ 𝑤 ↔ [𝑥 / 𝑤]𝑧 ∈ 𝑤) |
3 | nfv 1830 | . . . 4 ⊢ Ⅎ𝑤 𝑧 ∈ 𝑦 | |
4 | elequ2 1991 | . . . 4 ⊢ (𝑤 = 𝑦 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦)) | |
5 | 3, 4 | sbie 2396 | . . 3 ⊢ ([𝑦 / 𝑤]𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦) |
6 | 5 | sbbii 1874 | . 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑤]𝑧 ∈ 𝑤 ↔ [𝑥 / 𝑦]𝑧 ∈ 𝑦) |
7 | nfv 1830 | . . 3 ⊢ Ⅎ𝑤 𝑧 ∈ 𝑥 | |
8 | elequ2 1991 | . . 3 ⊢ (𝑤 = 𝑥 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑥)) | |
9 | 7, 8 | sbie 2396 | . 2 ⊢ ([𝑥 / 𝑤]𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑥) |
10 | 2, 6, 9 | 3bitr3i 289 | 1 ⊢ ([𝑥 / 𝑦]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 [wsb 1867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 |
This theorem is referenced by: nfnid 4823 |
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