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Theorem hbalw 2134
Description: Weak version of hbal 2192. Uses only Tarski's FOL axiom schemes. Unlike hbal 2192, this theorem requires that 𝑥 and 𝑦 be distinct, i.e. not be bundled. (Contributed by NM, 19-Apr-2017.)
Hypotheses
Ref Expression
hbalw.1 (𝑥 = 𝑧 → (𝜑𝜓))
hbalw.2 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
hbalw (∀𝑦𝜑 → ∀𝑥𝑦𝜑)
Distinct variable groups:   𝑥,𝑧   𝑥,𝑦   𝜑,𝑧   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦,𝑧)

Proof of Theorem hbalw
StepHypRef Expression
1 hbalw.2 . . 3 (𝜑 → ∀𝑥𝜑)
21alimi 1887 . 2 (∀𝑦𝜑 → ∀𝑦𝑥𝜑)
3 hbalw.1 . . 3 (𝑥 = 𝑧 → (𝜑𝜓))
43alcomiw 2127 . 2 (∀𝑦𝑥𝜑 → ∀𝑥𝑦𝜑)
52, 4syl 17 1 (∀𝑦𝜑 → ∀𝑥𝑦𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853
This theorem is referenced by: (None)
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