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Theorem wl-cbv3vv 33539
Description: Avoiding ax-11 2147. (Contributed by Wolf Lammen, 30-Aug-2021.)
Hypotheses
Ref Expression
wl-cbv3vv.nf 𝑥𝜓
wl-cbv3vv.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
wl-cbv3vv (∀𝑥𝜑 → ∀𝑦𝜓)
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)

Proof of Theorem wl-cbv3vv
StepHypRef Expression
1 wl-cbv3vv.nf . . 3 𝑥𝜓
2 wl-cbv3vv.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
31, 2spimv1 2226 . 2 (∀𝑥𝜑𝜓)
43alrimiv 1968 1 (∀𝑥𝜑 → ∀𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1594  wnf 1821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-12 2160
This theorem depends on definitions:  df-bi 197  df-ex 1818  df-nf 1823
This theorem is referenced by: (None)
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