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| Mirrors > Home > MPE Home > Th. List > axc16nfALT | Structured version Visualization version GIF version | ||
| Description: Alternate proof of axc16nf 2262, shorter but requiring ax-11 2156 and ax-13 2376. (Contributed by Mario Carneiro, 7-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| axc16nfALT | ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nfae 2437 | . 2 ⊢ Ⅎ𝑧∀𝑥 𝑥 = 𝑦 | |
| 2 | axc16g 2259 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑)) | |
| 3 | 1, 2 | nf5d 2283 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1537 Ⅎwnf 1782 | 
| 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-10 2140 ax-11 2156 ax-12 2176 ax-13 2376 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 | 
| This theorem is referenced by: (None) | 
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