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Mirrors > Home > MPE Home > Th. List > aev2 | Structured version Visualization version GIF version |
Description: A version of aev 2053
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 2052, aev 2053,
aev2 2054).
Using aev 2053 and alrimiv 1919, 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 28166. This list cannot be extended to ¬ or ⊥ since the scheme ∀𝑥𝑥 = 𝑦 is consistent with ax-mp 5, ax-gen 1787, ax-1 6-- ax-13 2381 (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 2053 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑤 𝑤 = 𝑠) | |
2 | aev 2053 | . . 3 ⊢ (∀𝑤 𝑤 = 𝑠 → ∀𝑡 𝑢 = 𝑣) | |
3 | 2 | alrimiv 1919 | . 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 1526 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 |
This theorem is referenced by: hbaev 2055 |
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