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Mirrors > Home > MPE Home > Th. List > aev2 | Structured version Visualization version GIF version |
Description: A version of aev 2058
with two universal quantifiers in the consequent.
One can prove similar statements with arbitrary numbers of universal
quantifiers in the consequent (the series begins with aeveq 2057, aev 2058,
aev2 2059).
Using aev 2058 and alrimiv 1924, one can actually prove (with no more axioms) any scheme of the form (∀𝑥𝑥 = 𝑦 → PHI) , DV (𝑥, 𝑦) where PHI involves only setvar variables and the connectors →, ↔, ∧, ∨, ⊤, =, ∀, ∃, ∃*, ∃!, Ⅎ. An example is given by aevdemo 28233. This list cannot be extended to ¬ or ⊥ since the scheme ∀𝑥𝑥 = 𝑦 is consistent with ax-mp 5, ax-gen 1792, ax-1 6-- ax-13 2386 (as the one-element universe shows). (Contributed by BJ, 23-Mar-2021.) |
Ref | Expression |
---|---|
aev2 | ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑡 𝑢 = 𝑣) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aev 2058 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑤 𝑤 = 𝑠) | |
2 | aev 2058 | . . 3 ⊢ (∀𝑤 𝑤 = 𝑠 → ∀𝑡 𝑢 = 𝑣) | |
3 | 2 | alrimiv 1924 | . 2 ⊢ (∀𝑤 𝑤 = 𝑠 → ∀𝑧∀𝑡 𝑢 = 𝑣) |
4 | 1, 3 | syl 17 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑡 𝑢 = 𝑣) |
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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1777 |
This theorem is referenced by: hbaev 2060 |
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