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Mirrors > Home > MPE Home > Th. List > Mathboxes > eqelb | Structured version Visualization version GIF version |
Description: Substitution of equal classes into elementhood relation. (Contributed by Peter Mazsa, 17-Jul-2019.) |
Ref | Expression |
---|---|
eqelb | ⊢ ((𝐴 = 𝐵 ∧ 𝐴 ∈ 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐵 ∈ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 483 | . . . 4 ⊢ ((𝐵 = 𝐴 ∧ 𝐴 ∈ 𝐶) → 𝐵 = 𝐴) | |
2 | eqeltr 35382 | . . . 4 ⊢ ((𝐵 = 𝐴 ∧ 𝐴 ∈ 𝐶) → 𝐵 ∈ 𝐶) | |
3 | 1, 2 | jca 512 | . . 3 ⊢ ((𝐵 = 𝐴 ∧ 𝐴 ∈ 𝐶) → (𝐵 = 𝐴 ∧ 𝐵 ∈ 𝐶)) |
4 | eqcom 2825 | . . . 4 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵) | |
5 | 4 | anbi1i 623 | . . 3 ⊢ ((𝐵 = 𝐴 ∧ 𝐴 ∈ 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐴 ∈ 𝐶)) |
6 | 4 | anbi1i 623 | . . 3 ⊢ ((𝐵 = 𝐴 ∧ 𝐵 ∈ 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐵 ∈ 𝐶)) |
7 | 3, 5, 6 | 3imtr3i 292 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 ∈ 𝐶) → (𝐴 = 𝐵 ∧ 𝐵 ∈ 𝐶)) |
8 | simpl 483 | . . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 = 𝐵) | |
9 | eqeltr 35382 | . . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶) | |
10 | 8, 9 | jca 512 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 ∈ 𝐶) → (𝐴 = 𝐵 ∧ 𝐴 ∈ 𝐶)) |
11 | 7, 10 | impbii 210 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 ∈ 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐵 ∈ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-cleq 2811 df-clel 2890 |
This theorem is referenced by: inxpxrn 35523 |
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