![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > equsalv | Structured version Visualization version GIF version |
Description: An equivalence related to implicit substitution. Version of equsal 2411 with a disjoint variable condition, which does not require ax-13 2366. See equsalvw 1999 for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsexv 2254. (Contributed by NM, 2-Jun-1993.) (Revised by BJ, 31-May-2019.) |
Ref | Expression |
---|---|
equsalv.nf | ⊢ Ⅎ𝑥𝜓 |
equsalv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
equsalv | ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equsalv.nf | . . 3 ⊢ Ⅎ𝑥𝜓 | |
2 | 1 | 19.23 2199 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ (∃𝑥 𝑥 = 𝑦 → 𝜓)) |
3 | equsalv.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 3 | pm5.74i 270 | . . 3 ⊢ ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜓)) |
5 | 4 | albii 1813 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜓)) |
6 | ax6ev 1965 | . . 3 ⊢ ∃𝑥 𝑥 = 𝑦 | |
7 | 6 | a1bi 361 | . 2 ⊢ (𝜓 ↔ (∃𝑥 𝑥 = 𝑦 → 𝜓)) |
8 | 2, 5, 7 | 3bitr4i 302 | 1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1531 ∃wex 1773 Ⅎwnf 1777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-12 2166 |
This theorem depends on definitions: df-bi 206 df-ex 1774 df-nf 1778 |
This theorem is referenced by: equsexv 2254 equsexvOLD 2255 equsalhw 2280 sbiev 2303 sb6rfv 2348 nfabdwOLD 2924 bj-equsalhv 36316 ichnfimlem 46832 |
Copyright terms: Public domain | W3C validator |