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Mirrors > Home > NFE Home > Th. List > nfeq | GIF version |
Description: Hypothesis builder for equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Ref | Expression |
---|---|
nfnfc.1 | ⊢ ℲxA |
nfeq.2 | ⊢ ℲxB |
Ref | Expression |
---|---|
nfeq | ⊢ Ⅎx A = B |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcleq 2347 | . 2 ⊢ (A = B ↔ ∀z(z ∈ A ↔ z ∈ B)) | |
2 | nfnfc.1 | . . . . 5 ⊢ ℲxA | |
3 | 2 | nfcri 2483 | . . . 4 ⊢ Ⅎx z ∈ A |
4 | nfeq.2 | . . . . 5 ⊢ ℲxB | |
5 | 4 | nfcri 2483 | . . . 4 ⊢ Ⅎx z ∈ B |
6 | 3, 5 | nfbi 1834 | . . 3 ⊢ Ⅎx(z ∈ A ↔ z ∈ B) |
7 | 6 | nfal 1842 | . 2 ⊢ Ⅎx∀z(z ∈ A ↔ z ∈ B) |
8 | 1, 7 | nfxfr 1570 | 1 ⊢ Ⅎx A = B |
Colors of variables: wff setvar class |
Syntax hints: ↔ 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-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-cleq 2346 df-clel 2349 df-nfc 2478 |
This theorem is referenced by: nfel 2497 nfeq1 2498 nfeq2 2500 nfne 2610 raleqf 2803 rexeqf 2804 reueq1f 2805 rmoeq1f 2806 rabeqf 2852 sbceqg 3152 csbhypf 3171 nffn 5180 nffo 5268 eqfnfv2f 5396 dff13f 5472 ov2gf 5711 fvmptf 5722 |
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