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Mirrors > Home > NFE Home > Th. List > ralab2 | 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 2619 | . 2 ⊢ (∀x ∈ {y ∣ φ}ψ ↔ ∀x(x ∈ {y ∣ φ} → ψ)) | |
2 | nfsab1 2343 | . . . 4 ⊢ Ⅎy x ∈ {y ∣ φ} | |
3 | nfv 1619 | . . . 4 ⊢ Ⅎyψ | |
4 | 2, 3 | nfim 1813 | . . 3 ⊢ Ⅎy(x ∈ {y ∣ φ} → ψ) |
5 | nfv 1619 | . . 3 ⊢ Ⅎx(φ → χ) | |
6 | eleq1 2413 | . . . . 5 ⊢ (x = y → (x ∈ {y ∣ φ} ↔ y ∈ {y ∣ φ})) | |
7 | abid 2341 | . . . . 5 ⊢ (y ∈ {y ∣ φ} ↔ φ) | |
8 | 6, 7 | syl6bb 252 | . . . 4 ⊢ (x = y → (x ∈ {y ∣ φ} ↔ φ)) |
9 | ralab2.1 | . . . 4 ⊢ (x = y → (ψ ↔ χ)) | |
10 | 8, 9 | imbi12d 311 | . . 3 ⊢ (x = y → ((x ∈ {y ∣ φ} → ψ) ↔ (φ → χ))) |
11 | 4, 5, 10 | cbval 1984 | . 2 ⊢ (∀x(x ∈ {y ∣ φ} → ψ) ↔ ∀y(φ → χ)) |
12 | 1, 11 | bitri 240 | 1 ⊢ (∀x ∈ {y ∣ φ}ψ ↔ ∀y(φ → χ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∀wal 1540 = wceq 1642 ∈ wcel 1710 {cab 2339 ∀wral 2614 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-ral 2619 |
This theorem is referenced by: ralrab2 3002 ssintab 3943 |
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