Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-eleq1w | Structured version Visualization version GIF version |
Description: Weaker version of eleq1 2676 (but more general than elequ1 1984) not depending on ax-ext 2590 (nor ax-12 2034 nor df-cleq 2603). Remark: this can also be done with eleq1i 2679, eqeltri 2684, eqeltrri 2685, eleq1a 2683, eleq1d 2672, eqeltrd 2688, eqeltrrd 2689, eqneltrd 2707, eqneltrrd 2708, nelneq 2712. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-eleq1w | ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equequ2 1940 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 ↔ 𝑧 = 𝑦)) | |
2 | 1 | anbi1d 737 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑧 = 𝑥 ∧ 𝑧 ∈ 𝐴) ↔ (𝑧 = 𝑦 ∧ 𝑧 ∈ 𝐴))) |
3 | 2 | exbidv 1837 | . 2 ⊢ (𝑥 = 𝑦 → (∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝐴) ↔ ∃𝑧(𝑧 = 𝑦 ∧ 𝑧 ∈ 𝐴))) |
4 | df-clel 2606 | . 2 ⊢ (𝑥 ∈ 𝐴 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝐴)) | |
5 | df-clel 2606 | . 2 ⊢ (𝑦 ∈ 𝐴 ↔ ∃𝑧(𝑧 = 𝑦 ∧ 𝑧 ∈ 𝐴)) | |
6 | 3, 4, 5 | 3bitr4g 302 | 1 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∃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 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-clel 2606 |
This theorem is referenced by: bj-clelsb3 32042 bj-nfcjust 32044 |
Copyright terms: Public domain | W3C validator |