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Mirrors > Home > ILE Home > Th. List > eleq2w | GIF version |
Description: Weaker version of eleq2 2241 (but more general than elequ2 2153) not depending on ax-ext 2159 nor df-cleq 2170. (Contributed by BJ, 29-Sep-2019.) |
Ref | Expression |
---|---|
eleq2w | ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elequ2 2153 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)) | |
2 | 1 | anbi2d 464 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑧 = 𝐴 ∧ 𝑧 ∈ 𝑥) ↔ (𝑧 = 𝐴 ∧ 𝑧 ∈ 𝑦))) |
3 | 2 | exbidv 1825 | . 2 ⊢ (𝑥 = 𝑦 → (∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝑥) ↔ ∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝑦))) |
4 | df-clel 2173 | . 2 ⊢ (𝐴 ∈ 𝑥 ↔ ∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝑥)) | |
5 | df-clel 2173 | . 2 ⊢ (𝐴 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝑦)) | |
6 | 3, 4, 5 | 3bitr4g 223 | 1 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∃wex 1492 ∈ wcel 2148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-14 2151 |
This theorem depends on definitions: df-bi 117 df-clel 2173 |
This theorem is referenced by: exmidontriimlem4 7217 |
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