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| Mirrors > Home > MPE Home > Th. List > axc16b | Structured version Visualization version GIF version | ||
| Description: This theorem shows that Axiom ax-c16 38893 is redundant in the presence of Theorem dtruALT2 5370, which states simply that at least two things exist. This justifies the remark at mmzfcnd.html#twoness 5370 (which links to this theorem). (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by NM, 7-Nov-2006.) |
| Ref | Expression |
|---|---|
| axc16b | ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dtruALT2 5370 | . 2 ⊢ ¬ ∀𝑥 𝑥 = 𝑦 | |
| 2 | 1 | pm2.21i 119 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-nul 5306 ax-pow 5365 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 |
| This theorem is referenced by: (None) |
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