| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > ralab2 | Structured version Visualization version GIF version | ||
| Description: Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) Drop ax-8 2110. (Revised by GG, 1-Dec-2023.) |
| Ref | Expression |
|---|---|
| ralab2.1 | ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) |
| Ref | Expression |
|---|---|
| ralab2 | ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∀𝑦(𝜑 → 𝜒)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-ral 3062 | . 2 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜓)) | |
| 2 | nfsab1 2722 | . . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ {𝑦 ∣ 𝜑} | |
| 3 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑦𝜓 | |
| 4 | 2, 3 | nfim 1896 | . . 3 ⊢ Ⅎ𝑦(𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜓) |
| 5 | nfv 1914 | . . 3 ⊢ Ⅎ𝑥(𝜑 → 𝜒) | |
| 6 | eleq1ab 2716 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ 𝑦 ∈ {𝑦 ∣ 𝜑})) | |
| 7 | abid 2718 | . . . . 5 ⊢ (𝑦 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑) | |
| 8 | 6, 7 | bitrdi 287 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑)) |
| 9 | ralab2.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) | |
| 10 | 8, 9 | imbi12d 344 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜓) ↔ (𝜑 → 𝜒))) |
| 11 | 4, 5, 10 | cbvalv1 2343 | . 2 ⊢ (∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜓) ↔ ∀𝑦(𝜑 → 𝜒)) |
| 12 | 1, 11 | bitri 275 | 1 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∀𝑦(𝜑 → 𝜒)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 ∈ wcel 2108 {cab 2714 ∀wral 3061 |
| 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-10 2141 ax-11 2157 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-ral 3062 |
| This theorem is referenced by: ralrab2 3704 elintabg 4957 ssintab 4965 efgval 19735 efger 19736 elintima 43666 |
| Copyright terms: Public domain | W3C validator |