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Mirrors > Home > MPE Home > Th. List > eqsb3 | Structured version Visualization version GIF version |
Description: Substitution applied to an atomic wff (class version of equsb3 2100). (Contributed by Rodolfo Medina, 28-Apr-2010.) |
Ref | Expression |
---|---|
eqsb3 | ⊢ ([𝑦 / 𝑥]𝑥 = 𝐴 ↔ 𝑦 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2822 | . 2 ⊢ (𝑥 = 𝑤 → (𝑥 = 𝐴 ↔ 𝑤 = 𝐴)) | |
2 | eqeq1 2822 | . 2 ⊢ (𝑤 = 𝑦 → (𝑤 = 𝐴 ↔ 𝑦 = 𝐴)) | |
3 | 1, 2 | sbievw2 2098 | 1 ⊢ ([𝑦 / 𝑥]𝑥 = 𝐴 ↔ 𝑦 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 [wsb 2060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-9 2115 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-sb 2061 df-cleq 2811 |
This theorem is referenced by: pm13.183 3656 pm13.183OLD 3657 eqsbc3 3814 eqsbc3OLD 3815 |
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