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Mirrors > Home > MPE Home > Th. List > nfeqd | Structured version Visualization version 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 2817 | . 2 ⊢ (𝐴 = 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) | |
2 | nfv 1915 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfeqd.1 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
4 | 3 | nfcrd 2971 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴) |
5 | nfeqd.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
6 | 5 | nfcrd 2971 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐵) |
7 | 4, 6 | nfbid 1903 | . . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) |
8 | 2, 7 | nfald 2347 | . 2 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) |
9 | 1, 8 | nfxfrd 1854 | 1 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 Ⅎwnfc 2963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 df-cleq 2816 df-nfc 2965 |
This theorem is referenced by: nfeld 2991 nfeq 2993 nfned 3122 vtoclgft 3555 vtoclgftOLD 3556 sbcralt 3857 csbiebt 3914 csbie2df 4394 dfnfc2 4862 eusvnfb 5296 eusv2i 5297 dfid3 5464 iota2df 6344 riotaeqimp 7142 riota5f 7144 oprabid 7190 axrepndlem1 10016 axrepndlem2 10017 axunnd 10020 axpowndlem4 10024 axregndlem2 10027 axinfndlem1 10029 axinfnd 10030 axacndlem4 10034 axacndlem5 10035 axacnd 10036 riotasv2d 36095 nfxnegd 41722 |
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