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.) |
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 2917 | . 2 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜓)) | |
2 | nfsab1 2612 | . . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ {𝑦 ∣ 𝜑} | |
3 | nfv 1843 | . . . 4 ⊢ Ⅎ𝑦𝜓 | |
4 | 2, 3 | nfim 1825 | . . 3 ⊢ Ⅎ𝑦(𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜓) |
5 | nfv 1843 | . . 3 ⊢ Ⅎ𝑥(𝜑 → 𝜒) | |
6 | eleq1 2689 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ 𝑦 ∈ {𝑦 ∣ 𝜑})) | |
7 | abid 2610 | . . . . 5 ⊢ (𝑦 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑) | |
8 | 6, 7 | syl6bb 276 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑)) |
9 | ralab2.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) | |
10 | 8, 9 | imbi12d 334 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜓) ↔ (𝜑 → 𝜒))) |
11 | 4, 5, 10 | cbval 2271 | . 2 ⊢ (∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜓) ↔ ∀𝑦(𝜑 → 𝜒)) |
12 | 1, 11 | bitri 264 | 1 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∀𝑦(𝜑 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1481 ∈ wcel 1990 {cab 2608 ∀wral 2912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-ral 2917 |
This theorem is referenced by: ralrab2 3372 ssintab 4494 efgval 18130 efger 18131 elintabg 37880 elintima 37945 |
Copyright terms: Public domain | W3C validator |