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Mirrors > Home > ILE Home > Th. List > sbid2h | GIF version |
Description: An identity law for substitution. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
sbid2h.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
Ref | Expression |
---|---|
sbid2h | ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbid2h.1 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | 1 | sbcof2 1739 | . 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
3 | 1 | sbh 1707 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑) |
4 | 2, 3 | bitri 183 | 1 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1288 [wsb 1693 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1382 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-11 1443 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 |
This theorem depends on definitions: df-bi 116 df-sb 1694 |
This theorem is referenced by: sbid2 1779 sb5rf 1781 sb6rf 1782 sbid2v 1921 |
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