Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > rabbid | Structured version Visualization version GIF version |
Description: Version of rabbidv 3477 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.) |
Ref | Expression |
---|---|
rabbid.n | ⊢ Ⅎ𝑥𝜑 |
rabbid.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
rabbid | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabbid.n | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | rabbid.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 2 | adantr 483 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) |
4 | 1, 3 | rabbida 3471 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1536 Ⅎwnf 1783 ∈ wcel 2113 {crab 3141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-ral 3142 df-rab 3146 |
This theorem is referenced by: satfv1 32631 bj-rabeqbid 34263 bj-seex 34265 |
Copyright terms: Public domain | W3C validator |