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Mirrors > Home > NFE Home > Th. List > nfcd | GIF version |
Description: Deduce that a class A does not have x free in it. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Ref | Expression |
---|---|
nfcd.1 | ⊢ Ⅎyφ |
nfcd.2 | ⊢ (φ → Ⅎx y ∈ A) |
Ref | Expression |
---|---|
nfcd | ⊢ (φ → ℲxA) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcd.1 | . . 3 ⊢ Ⅎyφ | |
2 | nfcd.2 | . . 3 ⊢ (φ → Ⅎx y ∈ A) | |
3 | 1, 2 | alrimi 1765 | . 2 ⊢ (φ → ∀yℲx y ∈ A) |
4 | df-nfc 2478 | . 2 ⊢ (ℲxA ↔ ∀yℲx y ∈ A) | |
5 | 3, 4 | sylibr 203 | 1 ⊢ (φ → ℲxA) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 Ⅎwnf 1544 ∈ wcel 1710 Ⅎwnfc 2476 |
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-11 1746 |
This theorem depends on definitions: df-bi 177 df-ex 1542 df-nf 1545 df-nfc 2478 |
This theorem is referenced by: nfabd2 2507 dvelimdc 2509 sbnfc2 3196 |
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