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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 2114. (Revised by Gino Giotto, 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 3056 | . 2 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜓)) | |
2 | nfsab1 2723 | . . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ {𝑦 ∣ 𝜑} | |
3 | nfv 1922 | . . . 4 ⊢ Ⅎ𝑦𝜓 | |
4 | 2, 3 | nfim 1904 | . . 3 ⊢ Ⅎ𝑦(𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜓) |
5 | nfv 1922 | . . 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 290 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑)) |
9 | ralab2.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) | |
10 | 8, 9 | imbi12d 348 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜓) ↔ (𝜑 → 𝜒))) |
11 | 4, 5, 10 | cbvalv1 2342 | . 2 ⊢ (∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜓) ↔ ∀𝑦(𝜑 → 𝜒)) |
12 | 1, 11 | bitri 278 | 1 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∀𝑦(𝜑 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1541 ∈ wcel 2112 {cab 2714 ∀wral 3051 |
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-10 2143 ax-11 2160 ax-12 2177 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-ral 3056 |
This theorem is referenced by: ralrab2 3601 ssintab 4862 efgval 19061 efger 19062 elintabg 40799 elintima 40879 |
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