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Mirrors > Home > MPE Home > Th. List > eleq2w2 | Structured version Visualization version GIF version |
Description: A weaker version of eleq2 2819 (but stronger than ax-9 2122 and elequ2 2127) that uses ax-12 2177 to avoid ax-8 2114 and df-clel 2809. Compare eleq2w 2814, 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 2729 | . . 3 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
2 | 1 | biimpi 219 | . 2 ⊢ (𝐴 = 𝐵 → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
3 | 2 | 19.21bi 2188 | 1 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1541 = wceq 1543 ∈ wcel 2112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-9 2122 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-cleq 2728 |
This theorem is referenced by: nfceqdf 2892 drnfc1 2916 drnfc2 2918 fineqvrep 32731 fineqvpow 32732 fineqvac 32733 f1omo 45804 |
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