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| Mirrors > Home > MPE Home > Th. List > elsb2 | Structured version Visualization version GIF version | ||
| Description: Substitution for the second argument of the non-logical predicate in an atomic formula. See elsb1 2116 for substitution for the first argument. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) Reduce axiom usage. (Revised by Wolf Lammen, 24-Jul-2023.) |
| Ref | Expression |
|---|---|
| elsb2 | ⊢ ([𝑦 / 𝑥]𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elequ2 2123 | . 2 ⊢ (𝑥 = 𝑤 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑤)) | |
| 2 | elequ2 2123 | . 2 ⊢ (𝑤 = 𝑦 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦)) | |
| 3 | 1, 2 | sbievw2 2098 | 1 ⊢ ([𝑦 / 𝑥]𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 [wsb 2064 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 |
| This theorem is referenced by: sbralie 3358 nfnid 5375 |
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