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Mirrors > Home > MPE Home > Th. List > clel2gOLD | Structured version Visualization version GIF version |
Description: Obsolete version of clel2g 3556 as of 1-Sep-2024. (Contributed by NM, 18-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
clel2gOLD | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1922 | . . 3 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 | |
2 | eleq1 2818 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
3 | 1, 2 | ceqsalg 3430 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵) ↔ 𝐴 ∈ 𝐵)) |
4 | 3 | bicomd 226 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∀𝑥(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-8 2114 ax-9 2122 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-3an 1091 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 |
This theorem is referenced by: (None) |
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