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Mirrors > Home > MPE Home > Th. List > nfcriiOLD | Structured version Visualization version GIF version |
Description: Obsolete version of nfcrii 2892 as of 23-May-2024. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfcriOLD.1 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nfcriiOLD | ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcriOLD.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
2 | 1 | nfcri 2887 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴 |
3 | 2 | nfsbv 2329 | . . 3 ⊢ Ⅎ𝑥[𝑦 / 𝑧]𝑧 ∈ 𝐴 |
4 | 3 | nf5ri 2193 | . 2 ⊢ ([𝑦 / 𝑧]𝑧 ∈ 𝐴 → ∀𝑥[𝑦 / 𝑧]𝑧 ∈ 𝐴) |
5 | clelsb3 2861 | . 2 ⊢ ([𝑦 / 𝑧]𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) | |
6 | 5 | albii 1827 | . 2 ⊢ (∀𝑥[𝑦 / 𝑧]𝑧 ∈ 𝐴 ↔ ∀𝑥 𝑦 ∈ 𝐴) |
7 | 4, 5, 6 | 3imtr3i 294 | 1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1541 [wsb 2070 ∈ wcel 2110 Ⅎwnfc 2880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-10 2141 ax-11 2158 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-nf 1792 df-sb 2071 df-clel 2812 df-nfc 2882 |
This theorem is referenced by: (None) |
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