Theorem bj-ax12w 34005

Description: The general statement that ax12w 2133 proves. (Contributed by BJ, 20-Mar-2020.)

Hypotheses

Ref	Expression
bj-ax12w.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))
bj-ax12w.2	⊢ (𝑦 = 𝑧 → (𝜓 ↔ 𝜃))

Assertion

Ref	Expression
bj-ax12w	⊢ (𝜑 → (∀𝑦𝜓 → ∀𝑥(𝜑 → 𝜓)))

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

Step	Hyp	Ref	Expression
1		bj-ax12w.2	. . 3 ⊢ (𝑦 = 𝑧 → (𝜓 ↔ 𝜃))
2	1	spw 2037	. 2 ⊢ (∀𝑦𝜓 → 𝜓)
3		bj-ax12w.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
4	3	bj-ax12wlem 33972	. 2 ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓)))
5	2, 4	syl5 34	1 ⊢ (𝜑 → (∀𝑦𝜓 → ∀𝑥(𝜑 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777

This theorem is referenced by: (None)