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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 36022 is redundant in the presence of theorem dtru 5263, which states simply that at least two things exist. This justifies the remark at mmzfcnd.html#twoness 5263 (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 | dtru 5263 | . 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 1531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-12 2173 ax-nul 5202 ax-pow 5258 |
This theorem depends on definitions: df-bi 209 df-an 399 df-tru 1536 df-ex 1777 df-nf 1781 |
This theorem is referenced by: (None) |
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