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Mirrors > Home > MPE Home > Th. List > sbelx | Structured version Visualization version GIF version |
Description: Elimination of substitution. Also see sbel2x 2421. (Contributed by NM, 5-Aug-1993.) Avoid ax-13 2301. (Revised by Wolf Lammen, 6-Aug-2023.) Avoid ax-10 2079. (Revised by Gino Giotto, 20-Aug-2023.) |
Ref | Expression |
---|---|
sbelx | ⊢ (𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑦]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbequ12r 2180 | . . 3 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑)) | |
2 | 1 | equsexvw 1962 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑦]𝜑) ↔ 𝜑) |
3 | 2 | bicomi 216 | 1 ⊢ (𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑦]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 387 ∃wex 1742 [wsb 2015 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-12 2106 |
This theorem depends on definitions: df-bi 199 df-an 388 df-ex 1743 df-sb 2016 |
This theorem is referenced by: pm13.196a 40163 |
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