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Mirrors > Home > MPE Home > Th. List > sbelx | Structured version Visualization version GIF version |
Description: Elimination of substitution. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
sbelx | ⊢ (𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑦]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbid2v 2445 | . 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ 𝜑) | |
2 | sb5 2418 | . 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑦]𝜑)) | |
3 | 1, 2 | bitr3i 265 | 1 ⊢ (𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑦]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∃wex 1695 [wsb 1867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-12 2034 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 |
This theorem is referenced by: pm13.196a 37637 |
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