Theorem bj-axc16g16 37679

Description: Proof of axc16g 2294 from { ax-1 6-- ax-7 2040, axc16 2295 }. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-axc16g16	⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧)

Proof of Theorem bj-axc16g16

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		aevlem 2089	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑡)
2		axc16 2295	. 2 ⊢ (∀𝑧 𝑧 = 𝑡 → (𝜑 → ∀𝑧𝜑))
3	1, 2	syl 18	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))

This proof depends on syntax axioms:   → wi 4  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812

This theorem is used by: (None)