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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 3477 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) |
3 | rabeqbidva.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | 3 | rabeqdv 3483 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜒} = {𝑥 ∈ 𝐵 ∣ 𝜒}) |
5 | 2, 4 | eqtrd 2854 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1531 ∈ wcel 2108 {crab 3140 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-ext 2791 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1775 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-rab 3145 |
This theorem is referenced by: natpropd 17238 gsumpropd2lem 17881 elntg 26762 rmfsupp2 30859 poimirlem28 34912 scotteqd 40563 domnmsuppn0 44407 eenglngeehlnm 44716 |
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