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Mirrors > Home > ILE 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 2131 | . 2 ⊢ (𝐴 = 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) | |
2 | nfv 1508 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfeqd.1 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
4 | 3 | nfcrd 2293 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴) |
5 | nfeqd.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
6 | 5 | nfcrd 2293 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐵) |
7 | 4, 6 | nfbid 1567 | . . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) |
8 | 2, 7 | nfald 1733 | . 2 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) |
9 | 1, 8 | nfxfrd 1451 | 1 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1329 = wceq 1331 Ⅎwnf 1436 ∈ wcel 1480 Ⅎwnfc 2266 |
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-5 1423 ax-7 1424 ax-gen 1425 ax-4 1487 ax-17 1506 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-cleq 2130 df-nfc 2268 |
This theorem is referenced by: nfeld 2295 nfned 2400 vtoclgft 2731 sbcralt 2980 sbcrext 2981 csbiebt 3034 dfnfc2 3749 eusvnfb 4370 eusv2i 4371 iota2df 5107 riota5f 5747 |
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