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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-reabeq | Structured version Visualization version GIF version |
Description: Relative form of abeq2 2944. (Contributed by BJ, 27-Dec-2023.) |
Ref | Expression |
---|---|
bj-reabeq | ⊢ ((𝑉 ∩ 𝐴) = {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrab3 4271 | . . 3 ⊢ {𝑥 ∈ 𝑉 ∣ 𝜑} = (𝑉 ∩ {𝑥 ∣ 𝜑}) | |
2 | 1 | eqeq2i 2833 | . 2 ⊢ ((𝑉 ∩ 𝐴) = {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ (𝑉 ∩ 𝐴) = (𝑉 ∩ {𝑥 ∣ 𝜑})) |
3 | nfcv 2976 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
4 | nfab1 2978 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
5 | nfcv 2976 | . . 3 ⊢ Ⅎ𝑥𝑉 | |
6 | 3, 4, 5 | bj-rcleqf 34359 | . 2 ⊢ ((𝑉 ∩ 𝐴) = (𝑉 ∩ {𝑥 ∣ 𝜑}) ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑})) |
7 | abid 2802 | . . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
8 | 7 | bibi2i 340 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}) ↔ (𝑥 ∈ 𝐴 ↔ 𝜑)) |
9 | 8 | ralbii 3164 | . 2 ⊢ (∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}) ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝜑)) |
10 | 2, 6, 9 | 3bitri 299 | 1 ⊢ ((𝑉 ∩ 𝐴) = {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1536 ∈ wcel 2113 {cab 2798 ∀wral 3137 {crab 3141 ∩ cin 3928 |
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-8 2115 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-clel 2892 df-nfc 2962 df-ral 3142 df-rab 3146 df-v 3493 df-in 3936 |
This theorem is referenced by: (None) |
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