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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-rcleqf | Structured version Visualization version GIF version |
Description: Relative version of cleqf 2931. (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 3869 | . . . . 5 ⊢ (𝑥 ∈ (𝑉 ∩ 𝐴) ↔ (𝑥 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴)) | |
2 | elin 3869 | . . . . 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 1826 | . 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 4117 | . . 3 ⊢ Ⅎ𝑥(𝑉 ∩ 𝐴) |
10 | bj-rcleqf.b | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
11 | 7, 10 | nfin 4117 | . . 3 ⊢ Ⅎ𝑥(𝑉 ∩ 𝐵) |
12 | 9, 11 | cleqf 2931 | . 2 ⊢ ((𝑉 ∩ 𝐴) = (𝑉 ∩ 𝐵) ↔ ∀𝑥(𝑥 ∈ (𝑉 ∩ 𝐴) ↔ 𝑥 ∈ (𝑉 ∩ 𝐵))) |
13 | df-ral 3059 | . 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 1540 = wceq 1542 ∈ wcel 2114 Ⅎwnfc 2880 ∀wral 3054 ∩ cin 3852 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ral 3059 df-rab 3063 df-v 3402 df-in 3860 |
This theorem is referenced by: bj-rcleq 34871 bj-reabeq 34872 |
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