Theorem bj-axc16g16 34013

Description: Proof of axc16g 2257 from { ax-1 6-- ax-7 2011, axc16 2258 }. (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 2056	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑡)
2		axc16 2258	. 2 ⊢ (∀𝑧 𝑧 = 𝑡 → (𝜑 → ∀𝑧𝜑))
3	1, 2	syl 17	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))

Syntax hints:   → wi 4  ∀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  ax-12 2173

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

This theorem is referenced by: (None)