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Mirrors > Home > MPE Home > Th. List > rabid2OLD | Structured version Visualization version GIF version |
Description: Obsolete version of rabid2 3463 as of 24-11-2024. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
rabid2OLD | ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqabb 2872 | . . 3 ⊢ (𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))) | |
2 | pm4.71 557 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))) | |
3 | 2 | albii 1820 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))) |
4 | 1, 3 | bitr4i 277 | . 2 ⊢ (𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) |
5 | df-rab 3432 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
6 | 5 | eqeq2i 2744 | . 2 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ 𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}) |
7 | df-ral 3061 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
8 | 4, 6, 7 | 3bitr4i 302 | 1 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1538 = wceq 1540 ∈ wcel 2105 {cab 2708 ∀wral 3060 {crab 3431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1543 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rab 3432 |
This theorem is referenced by: (None) |
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