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Mirrors > Home > NFE Home > Th. List > nfceqdf | GIF version |
Description: An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.) |
Ref | Expression |
---|---|
nfceqdf.1 | ⊢ Ⅎxφ |
nfceqdf.2 | ⊢ (φ → A = B) |
Ref | Expression |
---|---|
nfceqdf | ⊢ (φ → (ℲxA ↔ ℲxB)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfceqdf.1 | . . . 4 ⊢ Ⅎxφ | |
2 | nfceqdf.2 | . . . . 5 ⊢ (φ → A = B) | |
3 | 2 | eleq2d 2420 | . . . 4 ⊢ (φ → (y ∈ A ↔ y ∈ B)) |
4 | 1, 3 | nfbidf 1774 | . . 3 ⊢ (φ → (Ⅎx y ∈ A ↔ Ⅎx y ∈ B)) |
5 | 4 | albidv 1625 | . 2 ⊢ (φ → (∀yℲx y ∈ A ↔ ∀yℲx y ∈ B)) |
6 | df-nfc 2478 | . 2 ⊢ (ℲxA ↔ ∀yℲx y ∈ A) | |
7 | df-nfc 2478 | . 2 ⊢ (ℲxB ↔ ∀yℲx y ∈ B) | |
8 | 5, 6, 7 | 3bitr4g 279 | 1 ⊢ (φ → (ℲxA ↔ ℲxB)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∀wal 1540 Ⅎwnf 1544 = wceq 1642 ∈ 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 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-an 360 df-ex 1542 df-nf 1545 df-cleq 2346 df-clel 2349 df-nfc 2478 |
This theorem is referenced by: dfnfc2 3909 nfopd 4605 nfimad 4954 nffvd 5335 |
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