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Mirrors > Home > MPE Home > Th. List > nfccdeq | Structured version Visualization version GIF version |
Description: Variation of nfcdeq 3786 for classes. Usage of this theorem is discouraged because it depends on ax-13 2375. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-11 2155. (Revised by GG, 19-May-2023.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfccdeq.1 | ⊢ Ⅎ𝑥𝐴 |
nfccdeq.2 | ⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
nfccdeq | ⊢ 𝐴 = 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfccdeq.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | 1 | nfcri 2895 | . . 3 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴 |
3 | eqid 2735 | . . . . 5 ⊢ 𝑧 = 𝑧 | |
4 | 3 | cdeqth 3776 | . . . 4 ⊢ CondEq(𝑥 = 𝑦 → 𝑧 = 𝑧) |
5 | nfccdeq.2 | . . . 4 ⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵) | |
6 | 4, 5 | cdeqel 3785 | . . 3 ⊢ CondEq(𝑥 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵)) |
7 | 2, 6 | nfcdeq 3786 | . 2 ⊢ (𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵) |
8 | 7 | eqriv 2732 | 1 ⊢ 𝐴 = 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 Ⅎwnfc 2888 CondEqwcdeq 3772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-13 2375 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1777 df-nf 1781 df-sb 2063 df-cleq 2727 df-clel 2814 df-nfc 2890 df-cdeq 3773 |
This theorem is referenced by: (None) |
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