![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > rabeqbidva | Structured version Visualization version GIF version |
Description: Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.) |
Ref | Expression |
---|---|
rabeqbidva.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
rabeqbidva.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
rabeqbidva | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabeqbidva.2 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) | |
2 | 1 | rabbidva 3425 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) |
3 | rabeqbidva.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | 3 | rabeqdv 3432 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜒} = {𝑥 ∈ 𝐵 ∣ 𝜒}) |
5 | 2, 4 | eqtrd 2833 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {crab 3110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-rab 3115 |
This theorem is referenced by: natpropd 17238 gsumpropd2lem 17881 elntg 26778 rmfsupp2 30917 poimirlem28 35085 scotteqd 40945 domnmsuppn0 44771 eenglngeehlnm 45153 |
Copyright terms: Public domain | W3C validator |