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Mirrors > Home > MPE Home > Th. List > elintabOLD | Structured version Visualization version GIF version |
Description: Obsolete version of elintab 4963 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 4957 | . 2 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ 𝑦)) |
3 | nfsab1 2720 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} | |
4 | nfv 1912 | . . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑦 | |
5 | 3, 4 | nfim 1894 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ 𝑦) |
6 | nfv 1912 | . . 3 ⊢ Ⅎ𝑦(𝜑 → 𝐴 ∈ 𝑥) | |
7 | eleq1w 2822 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑥 ∣ 𝜑})) | |
8 | abid 2716 | . . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
9 | 7, 8 | bitrdi 287 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)) |
10 | eleq2w 2823 | . . . 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 2342 | . 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 1535 ∈ wcel 2106 {cab 2712 Vcvv 3478 ∩ cint 4951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-int 4952 |
This theorem is referenced by: (None) |
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