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Theorem wl-ax11-lem10 34825
Description: We now have prepared everything. The unwanted variable 𝑢 is just in one place left. pm2.61 194 can be used in conjunction with wl-ax11-lem9 34824 to eliminate the second antecedent. Missing is something along the lines of ax-6 1966, so we could remove the first antecedent. But the Metamath axioms cannot accomplish this. Such a rule must reside one abstraction level higher than all others: It says that a distinctor implies a distinct variable condition on its contained setvar. This is only needed if such conditions are required, as ax-11v does. The result of this study is for me, that you cannot introduce a setvar capturing this condition, and hope to eliminate it later. (Contributed by Wolf Lammen, 30-Jun-2019.)
Assertion
Ref Expression
wl-ax11-lem10 (∀𝑦 𝑦 = 𝑢 → (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑦𝑥𝜑 → ∀𝑥𝑦𝜑)))
Distinct variable group:   𝑥,𝑢
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑢)

Proof of Theorem wl-ax11-lem10
StepHypRef Expression
1 wl-ax11-lem8 34823 . . . . 5 ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑢𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝑥𝜑))
2 wl-ax11-lem6 34821 . . . . 5 ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑢𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑥𝑦𝜑))
31, 2bitr3d 283 . . . 4 ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑦𝑥𝜑 ↔ ∀𝑥𝑦𝜑))
43biimpd 231 . . 3 ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑦𝑥𝜑 → ∀𝑥𝑦𝜑))
54ex 415 . 2 (∀𝑢 𝑢 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑦𝑥𝜑 → ∀𝑥𝑦𝜑)))
65aecoms 2446 1 (∀𝑦 𝑦 = 𝑢 → (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑦𝑥𝜑 → ∀𝑥𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398  wal 1531  [wsb 2065
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-10 2141  ax-12 2173  ax-13 2386  ax-wl-11v 34815
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066
This theorem is referenced by: (None)
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