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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 2114 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 2121 | . 2 ⊢ (𝑥 = 𝑤 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑤)) | |
2 | elequ2 2121 | . 2 ⊢ (𝑤 = 𝑦 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦)) | |
3 | 1, 2 | sbievw2 2096 | 1 ⊢ ([𝑦 / 𝑥]𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 [wsb 2062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-9 2116 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-sb 2063 |
This theorem is referenced by: sbralie 3356 nfnid 5381 |
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