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Mirrors > Home > ILE 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 2158 | . 2 ⊢ (𝐴 = 𝐵 ↔ ∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵)) | |
2 | nfnfc.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfcri 2300 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴 |
4 | nfeq.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
5 | 4 | nfcri 2300 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐵 |
6 | 3, 5 | nfbi 1576 | . . 3 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵) |
7 | 6 | nfal 1563 | . 2 ⊢ Ⅎ𝑥∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵) |
8 | 1, 7 | nfxfr 1461 | 1 ⊢ Ⅎ𝑥 𝐴 = 𝐵 |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∀wal 1340 = wceq 1342 Ⅎwnf 1447 ∈ wcel 2135 Ⅎwnfc 2293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-cleq 2157 df-clel 2160 df-nfc 2295 |
This theorem is referenced by: nfel 2315 nfeq1 2316 nfeq2 2318 nfne 2427 raleqf 2655 rexeqf 2656 reueq1f 2657 rmoeq1f 2658 rabeqf 2711 sbceqg 3056 csbhypf 3078 nfiotadw 5150 nffn 5278 nffo 5403 fvmptdf 5567 mpteqb 5570 fvmptf 5572 eqfnfv2f 5581 dff13f 5732 ovmpos 5956 ov2gf 5957 ovmpodxf 5958 ovmpodf 5964 eqerlem 6523 sumeq2 11286 fsumadd 11333 prodeq1f 11479 prodeq2 11484 txcnp 12812 cnmpt11 12824 cnmpt21 12832 cnmptcom 12839 |
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