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Mirrors > Home > MPE Home > Th. List > sb6rf | Structured version Visualization version GIF version |
Description: Reversed substitution. For a version requiring disjoint variables, but fewer axioms, see sb6rfv 2380. (Contributed by NM, 1-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 21-Sep-2018.) |
Ref | Expression |
---|---|
sb5rf.1 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
sb6rf | ⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb5rf.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
2 | sbequ12r 2287 | . . 3 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑)) | |
3 | 1, 2 | equsal 2437 | . 2 ⊢ (∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑) ↔ 𝜑) |
4 | 3 | bicomi 216 | 1 ⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∀wal 1654 Ⅎwnf 1882 [wsb 2067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-12 2220 ax-13 2389 |
This theorem depends on definitions: df-bi 199 df-an 387 df-ex 1879 df-nf 1883 df-sb 2068 |
This theorem is referenced by: eu1OLD 2696 |
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