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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-rabeqbida | Structured version Visualization version GIF version |
Description: Version of rabeqbidva 3196 with two dv conditions removed and the third replaced by a non-freeness hypothesis. (Contributed by BJ, 27-Apr-2019.) |
Ref | Expression |
---|---|
bj-rabeqbida.nf | ⊢ Ⅎ𝑥𝜑 |
bj-rabeqbida.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
bj-rabeqbida.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
bj-rabeqbida | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-rabeqbida.nf | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | bj-rabeqbida.2 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) | |
3 | 1, 2 | bj-rabbida 32914 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) |
4 | bj-rabeqbida.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
5 | 1, 4 | bj-rabeqd 32916 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜒} = {𝑥 ∈ 𝐵 ∣ 𝜒}) |
6 | 3, 5 | eqtrd 2656 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 {crab 2916 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-ral 2917 df-rab 2921 |
This theorem is referenced by: (None) |
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