![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > elsb1 | Structured version Visualization version GIF version |
Description: Substitution for the first argument of the non-logical predicate in an atomic formula. See elsb2 2115 for substitution for the second argument. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) Reduce axiom usage. (Revised by Wolf Lammen, 24-Jul-2023.) |
Ref | Expression |
---|---|
elsb1 | ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elequ1 2105 | . 2 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ 𝑧 ↔ 𝑤 ∈ 𝑧)) | |
2 | elequ1 2105 | . 2 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)) | |
3 | 1, 2 | sbievw2 2091 | 1 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 [wsb 2059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-sb 2060 |
This theorem is referenced by: cvjust 2718 |
Copyright terms: Public domain | W3C validator |