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Mirrors > Home > NFE Home > Th. List > nfeqd | GIF version |
Description: Hypothesis builder for equality. (Contributed by Mario Carneiro, 7-Oct-2016.) |
Ref | Expression |
---|---|
nfeqd.1 | ⊢ (φ → ℲxA) |
nfeqd.2 | ⊢ (φ → ℲxB) |
Ref | Expression |
---|---|
nfeqd | ⊢ (φ → Ⅎx A = B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcleq 2347 | . 2 ⊢ (A = B ↔ ∀y(y ∈ A ↔ y ∈ B)) | |
2 | nfv 1619 | . . 3 ⊢ Ⅎyφ | |
3 | nfeqd.1 | . . . . 5 ⊢ (φ → ℲxA) | |
4 | 3 | nfcrd 2502 | . . . 4 ⊢ (φ → Ⅎx y ∈ A) |
5 | nfeqd.2 | . . . . 5 ⊢ (φ → ℲxB) | |
6 | 5 | nfcrd 2502 | . . . 4 ⊢ (φ → Ⅎx y ∈ B) |
7 | 4, 6 | nfbid 1832 | . . 3 ⊢ (φ → Ⅎx(y ∈ A ↔ y ∈ B)) |
8 | 2, 7 | nfald 1852 | . 2 ⊢ (φ → Ⅎx∀y(y ∈ A ↔ y ∈ B)) |
9 | 1, 8 | nfxfrd 1571 | 1 ⊢ (φ → Ⅎx A = B) |
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-6 1729 ax-7 1734 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-nfc 2478 |
This theorem is referenced by: nfeld 2504 nfned 2612 vtoclgft 2905 sbcralt 3118 csbiebt 3172 dfnfc2 3909 nfiotad 4342 iota2df 4365 dfid3 4768 oprabid 5550 |
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