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Mirrors > Home > MPE Home > Th. List > eqsb1 | Structured version Visualization version GIF version |
Description: Substitution for the left-hand side in an equality. Class version of equsb3 2101. (Contributed by Rodolfo Medina, 28-Apr-2010.) |
Ref | Expression |
---|---|
eqsb1 | ⊢ ([𝑦 / 𝑥]𝑥 = 𝐴 ↔ 𝑦 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2739 | . 2 ⊢ (𝑥 = 𝑤 → (𝑥 = 𝐴 ↔ 𝑤 = 𝐴)) | |
2 | eqeq1 2739 | . 2 ⊢ (𝑤 = 𝑦 → (𝑤 = 𝐴 ↔ 𝑦 = 𝐴)) | |
3 | 1, 2 | sbievw2 2096 | 1 ⊢ ([𝑦 / 𝑥]𝑥 = 𝐴 ↔ 𝑦 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 [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 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-sb 2063 df-cleq 2727 |
This theorem is referenced by: sbhypf 3544 pm13.183 3666 eqsbc1 3841 |
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