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| Mirrors > Home > MPE Home > Th. List > eleq2w2 | Structured version Visualization version GIF version | ||
| Description: A weaker version of eleq2 2830 (but stronger than ax-9 2118 and elequ2 2123) that uses ax-12 2177 to avoid ax-8 2110 and df-clel 2816. Compare eleq2w 2825, whose setvars appear where the class variables are in this theorem, and vice versa. (Contributed by BJ, 24-Jun-2019.) Strengthen from setvar variables to class variables. (Revised by WL and SN, 23-Aug-2024.) |
| Ref | Expression |
|---|---|
| eleq2w2 | ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfcleq 2730 | . . 3 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
| 2 | 1 | biimpi 216 | . 2 ⊢ (𝐴 = 𝐵 → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
| 3 | 2 | 19.21bi 2189 | 1 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 = wceq 1540 ∈ wcel 2108 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 |
| This theorem is referenced by: nfceqdf 2901 drnfc1 2925 drnfc2 2926 fineqvrep 35109 fineqvpow 35110 fineqvac 35111 fvineqsneu 37412 sge0f1o 46397 f1omo 48792 |
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