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Theorem bj-ax12w 32790
Description: The general statement that ax12w 2050 proves. (Contributed by BJ, 20-Mar-2020.)
Hypotheses
Ref Expression
bj-ax12w.1 (𝜑 → (𝜓𝜒))
bj-ax12w.2 (𝑦 = 𝑧 → (𝜓𝜃))
Assertion
Ref Expression
bj-ax12w (𝜑 → (∀𝑦𝜓 → ∀𝑥(𝜑𝜓)))
Distinct variable groups:   𝜒,𝑥   𝜃,𝑦   𝜓,𝑧   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦)   𝜒(𝑦,𝑧)   𝜃(𝑥,𝑧)

Proof of Theorem bj-ax12w
StepHypRef Expression
1 bj-ax12w.2 . . 3 (𝑦 = 𝑧 → (𝜓𝜃))
21spw 2009 . 2 (∀𝑦𝜓𝜓)
3 bj-ax12w.1 . . 3 (𝜑 → (𝜓𝜒))
43bj-ax12wlem 32742 . 2 (𝜑 → (𝜓 → ∀𝑥(𝜑𝜓)))
52, 4syl5 34 1 (𝜑 → (∀𝑦𝜓 → ∀𝑥(𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745
This theorem is referenced by: (None)
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