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| Description: Alternate proof of axc16g 2259 that uses df-sb 2064 and requires ax-10 2140, ax-11 2156, ax-13 2376. (Contributed by NM, 15-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| axc16gALT | ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | aev 2056 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑥) | |
| 2 | axc16ALT 2493 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑)) | |
| 3 | biidd 262 | . . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 → (𝜑 ↔ 𝜑)) | |
| 4 | 3 | dral1 2443 | . . 3 ⊢ (∀𝑧 𝑧 = 𝑥 → (∀𝑧𝜑 ↔ ∀𝑥𝜑)) | 
| 5 | 4 | biimprd 248 | . 2 ⊢ (∀𝑧 𝑧 = 𝑥 → (∀𝑥𝜑 → ∀𝑧𝜑)) | 
| 6 | 1, 2, 5 | sylsyld 61 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1537 | 
| 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-ex 1779 df-nf 1783 df-sb 2064 | 
| This theorem is referenced by: (None) | 
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