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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-eleq2w | Structured version Visualization version GIF version |
Description: Weaker version of eleq2 2677 (but more general than elequ2 1991) not depending on ax-ext 2590 (nor ax-12 2034 nor df-cleq 2603). (Contributed by BJ, 29-Sep-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-eleq2w | ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elequ2 1991 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)) | |
2 | 1 | anbi2d 736 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑧 = 𝐴 ∧ 𝑧 ∈ 𝑥) ↔ (𝑧 = 𝐴 ∧ 𝑧 ∈ 𝑦))) |
3 | 2 | exbidv 1837 | . 2 ⊢ (𝑥 = 𝑦 → (∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝑥) ↔ ∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝑦))) |
4 | df-clel 2606 | . 2 ⊢ (𝐴 ∈ 𝑥 ↔ ∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝑥)) | |
5 | df-clel 2606 | . 2 ⊢ (𝐴 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝑦)) | |
6 | 3, 4, 5 | 3bitr4g 302 | 1 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-clel 2606 |
This theorem is referenced by: (None) |
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