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Theorem ax5el 33023
Description: Theorem to add distinct quantifier to atomic formula. This theorem demonstrates the induction basis for ax-5 1826 considered as a metatheorem.) (Contributed by NM, 22-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax5el (𝑥𝑦 → ∀𝑧 𝑥𝑦)
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem ax5el
StepHypRef Expression
1 ax-c14 32977 . 2 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥𝑦 → ∀𝑧 𝑥𝑦)))
2 ax-c16 32978 . 2 (∀𝑧 𝑧 = 𝑥 → (𝑥𝑦 → ∀𝑧 𝑥𝑦))
3 ax-c16 32978 . 2 (∀𝑧 𝑧 = 𝑦 → (𝑥𝑦 → ∀𝑧 𝑥𝑦))
41, 2, 3pm2.61ii 175 1 (𝑥𝑦 → ∀𝑧 𝑥𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-c14 32977  ax-c16 32978
This theorem is referenced by:  dveel2ALT  33025
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