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Theorem ax11w 2162
Description: Weak version of ax-11 2190 from which we can prove any ax-11 2190 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. Unlike ax-11 2190, this theorem requires that 𝑥 and 𝑦 be distinct i.e. are not bundled. It is an alias of alcomiw 2127 introduced for labeling consistency. (Contributed by NM, 10-Apr-2017.) Use alcomiw 2127 instead. (New usage is discouraged.)
Hypothesis
Ref Expression
ax11w.1 (𝑦 = 𝑧 → (𝜑𝜓))
Assertion
Ref Expression
ax11w (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
Distinct variable groups:   𝑦,𝑧   𝑥,𝑦   𝜑,𝑧   𝜓,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑧)

Proof of Theorem ax11w
StepHypRef Expression
1 ax11w.1 . 2 (𝑦 = 𝑧 → (𝜑𝜓))
21alcomiw 2127 1 (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
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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