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| Description: A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2376. (Contributed by NM, 1-Jan-2002.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| nd2 | ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑧 ∈ 𝑦) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elirrv 9637 | . . 3 ⊢ ¬ 𝑧 ∈ 𝑧 | |
| 2 | stdpc4 2067 | . . . 4 ⊢ (∀𝑦 𝑧 ∈ 𝑦 → [𝑧 / 𝑦]𝑧 ∈ 𝑦) | |
| 3 | 1 | nfnth 1801 | . . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝑧 | 
| 4 | elequ2 2122 | . . . . 5 ⊢ (𝑦 = 𝑧 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑧)) | |
| 5 | 3, 4 | sbie 2506 | . . . 4 ⊢ ([𝑧 / 𝑦]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑧) | 
| 6 | 2, 5 | sylib 218 | . . 3 ⊢ (∀𝑦 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑧) | 
| 7 | 1, 6 | mto 197 | . 2 ⊢ ¬ ∀𝑦 𝑧 ∈ 𝑦 | 
| 8 | axc11 2434 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑧 ∈ 𝑦 → ∀𝑦 𝑧 ∈ 𝑦)) | |
| 9 | 7, 8 | mtoi 199 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑧 ∈ 𝑦) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∀wal 1537 [wsb 2063 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-12 2176 ax-13 2376 ax-ext 2707 ax-sep 5295 ax-pr 5431 ax-reg 9633 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-v 3481 df-un 3955 df-sn 4626 df-pr 4628 | 
| This theorem is referenced by: axrepnd 10635 axpownd 10642 axinfndlem1 10646 axacndlem4 10651 | 
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