Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > sb6rfv | Structured version Visualization version GIF version |
Description: Reversed substitution. Version of sb6rf 2491 requiring disjoint variables, but fewer axioms. (Contributed by NM, 1-Aug-1993.) (Revised by Wolf Lammen, 7-Feb-2023.) |
Ref | Expression |
---|---|
sb6rfv.nf | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
sb6rfv | ⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb6rfv.nf | . . 3 ⊢ Ⅎ𝑦𝜑 | |
2 | sbequ12r 2254 | . . 3 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑)) | |
3 | 1, 2 | equsalv 2268 | . 2 ⊢ (∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑) ↔ 𝜑) |
4 | 3 | bicomi 226 | 1 ⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 Ⅎwnf 1784 [wsb 2069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-nf 1785 df-sb 2070 |
This theorem is referenced by: eu1 2694 |
Copyright terms: Public domain | W3C validator |