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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 2616 | . 2 ⊢ (𝐴 = 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) | |
2 | nfv 1843 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfeqd.1 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
4 | 3 | nfcrd 2771 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴) |
5 | nfeqd.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
6 | 5 | nfcrd 2771 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐵) |
7 | 4, 6 | nfbid 1832 | . . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) |
8 | 2, 7 | nfald 2165 | . 2 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) |
9 | 1, 8 | nfxfrd 1780 | 1 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1481 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 Ⅎwnfc 2751 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ex 1705 df-nf 1710 df-cleq 2615 df-nfc 2753 |
This theorem is referenced by: nfeld 2773 nfeq 2776 nfned 2895 vtoclgft 3254 vtoclgftOLD 3255 sbcralt 3510 csbiebt 3553 dfnfc2 4454 dfnfc2OLD 4455 eusvnfb 4862 eusv2i 4863 dfid3 5025 iota2df 5875 riotaeqimp 6634 riota5f 6636 oprabid 6677 axrepndlem1 9414 axrepndlem2 9415 axunnd 9418 axpowndlem4 9422 axregndlem2 9425 axinfndlem1 9427 axinfnd 9428 axacndlem4 9432 axacndlem5 9433 axacnd 9434 riotasv2d 34243 nfxnegd 39668 |
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