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Mirrors > Home > MPE Home > Th. List > Mathboxes > rabimbieq | Structured version Visualization version GIF version |
Description: Restricted equivalent wff's correspond to restricted class abstractions which are equal with the same class. (Contributed by Peter Mazsa, 22-Jul-2021.) |
Ref | Expression |
---|---|
rabimbieq.1 | ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑} |
rabimbieq.2 | ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
rabimbieq | ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabimbieq.1 | . 2 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑} | |
2 | rabimbieq.2 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | rabbiia 3474 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓} |
4 | 1, 3 | eqtri 2846 | 1 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 {crab 3144 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-9 2124 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-sb 2070 df-clab 2802 df-cleq 2816 df-rab 3149 |
This theorem is referenced by: abeqinbi 35517 dfsymrels4 35785 dfsymrels5 35786 dffunsALTV2 35919 dffunsALTV3 35920 dffunsALTV4 35921 dfdisjs2 35944 dfdisjs5 35947 |
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