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| Mirrors > Home > MPE Home > Th. List > elintabOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of elintab 4958 as of 17-Jan-2025. (Contributed by NM, 30-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| elintab.ex | ⊢ 𝐴 ∈ V | 
| Ref | Expression | 
|---|---|
| elintabOLD | ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elintab.ex | . . 3 ⊢ 𝐴 ∈ V | |
| 2 | 1 | elint 4952 | . 2 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ 𝑦)) | 
| 3 | nfsab1 2722 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} | |
| 4 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑦 | |
| 5 | 3, 4 | nfim 1896 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ 𝑦) | 
| 6 | nfv 1914 | . . 3 ⊢ Ⅎ𝑦(𝜑 → 𝐴 ∈ 𝑥) | |
| 7 | eleq1w 2824 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑥 ∣ 𝜑})) | |
| 8 | abid 2718 | . . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
| 9 | 7, 8 | bitrdi 287 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)) | 
| 10 | eleq2w 2825 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑥)) | |
| 11 | 9, 10 | imbi12d 344 | . . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ 𝑦) ↔ (𝜑 → 𝐴 ∈ 𝑥))) | 
| 12 | 5, 6, 11 | cbvalv1 2343 | . 2 ⊢ (∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ 𝑦) ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)) | 
| 13 | 2, 12 | bitri 275 | 1 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 ∈ wcel 2108 {cab 2714 Vcvv 3480 ∩ cint 4946 | 
| 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-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-int 4947 | 
| This theorem is referenced by: (None) | 
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