Theorem axc16gb 2274

Description: Biconditional strengthening of axc16g 2272. (Contributed by NM, 15-May-1993.)

Assertion

Ref	Expression
axc16gb	⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ ∀𝑧𝜑))

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem axc16gb

Step	Hyp	Ref	Expression
1		axc16g 2272	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
2		sp 2195	. 2 ⊢ (∀𝑧𝜑 → 𝜑)
3	1, 2	impbid1 226	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ ∀𝑧𝜑))

Syntax hints:   → wi 4   ↔ wb 207  ∀wal 1545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787

This theorem is referenced by: (None)