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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-rcleqf | Structured version Visualization version GIF version |
Description: Relative version of cleqf 2983. (Contributed by BJ, 27-Dec-2023.) |
Ref | Expression |
---|---|
bj-rcleqf.a | ⊢ Ⅎ𝑥𝐴 |
bj-rcleqf.b | ⊢ Ⅎ𝑥𝐵 |
bj-rcleqf.v | ⊢ Ⅎ𝑥𝑉 |
Ref | Expression |
---|---|
bj-rcleqf | ⊢ ((𝑉 ∩ 𝐴) = (𝑉 ∩ 𝐵) ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3897 | . . . . 5 ⊢ (𝑥 ∈ (𝑉 ∩ 𝐴) ↔ (𝑥 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴)) | |
2 | elin 3897 | . . . . 5 ⊢ (𝑥 ∈ (𝑉 ∩ 𝐵) ↔ (𝑥 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵)) | |
3 | 1, 2 | bibi12i 343 | . . . 4 ⊢ ((𝑥 ∈ (𝑉 ∩ 𝐴) ↔ 𝑥 ∈ (𝑉 ∩ 𝐵)) ↔ ((𝑥 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵))) |
4 | pm5.32 577 | . . . 4 ⊢ ((𝑥 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) ↔ ((𝑥 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵))) | |
5 | 3, 4 | bitr4i 281 | . . 3 ⊢ ((𝑥 ∈ (𝑉 ∩ 𝐴) ↔ 𝑥 ∈ (𝑉 ∩ 𝐵)) ↔ (𝑥 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))) |
6 | 5 | albii 1821 | . 2 ⊢ (∀𝑥(𝑥 ∈ (𝑉 ∩ 𝐴) ↔ 𝑥 ∈ (𝑉 ∩ 𝐵)) ↔ ∀𝑥(𝑥 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))) |
7 | bj-rcleqf.v | . . . 4 ⊢ Ⅎ𝑥𝑉 | |
8 | bj-rcleqf.a | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
9 | 7, 8 | nfin 4143 | . . 3 ⊢ Ⅎ𝑥(𝑉 ∩ 𝐴) |
10 | bj-rcleqf.b | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
11 | 7, 10 | nfin 4143 | . . 3 ⊢ Ⅎ𝑥(𝑉 ∩ 𝐵) |
12 | 9, 11 | cleqf 2983 | . 2 ⊢ ((𝑉 ∩ 𝐴) = (𝑉 ∩ 𝐵) ↔ ∀𝑥(𝑥 ∈ (𝑉 ∩ 𝐴) ↔ 𝑥 ∈ (𝑉 ∩ 𝐵))) |
13 | df-ral 3111 | . 2 ⊢ (∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))) | |
14 | 6, 12, 13 | 3bitr4i 306 | 1 ⊢ ((𝑉 ∩ 𝐴) = (𝑉 ∩ 𝐵) ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1536 = wceq 1538 ∈ wcel 2111 Ⅎwnfc 2936 ∀wral 3106 ∩ cin 3880 |
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-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rab 3115 df-v 3443 df-in 3888 |
This theorem is referenced by: bj-rcleq 34462 bj-reabeq 34463 |
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