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Theorem bj-ax12w 31852
 Description: The general statement that ax12w 1997 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 1954 . 2 (∀𝑦𝜓𝜓)
3 bj-ax12w.1 . . 3 (𝜑 → (𝜓𝜒))
43bj-ax12wlem 31807 . 2 (𝜑 → (𝜓 → ∀𝑥(𝜑𝜓)))
52, 4syl5 33 1 (𝜑 → (∀𝑦𝜓 → ∀𝑥(𝜑𝜓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696 This theorem is referenced by: (None)
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