Theorem axc16gOLD 2161

Description: Obsolete proof of axc16g 2134 as of 11-Oct-2021. (Contributed by NM, 15-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 18-Feb-2018.) Remove dependency on ax-13 2246, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Revised by BJ, 7-Jul-2021.) Shorten axc11rv 2139. (Revised by Wolf Lammen, 11-Oct-2021.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
axc16gOLD	⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem axc16gOLD

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		aevlem 1981	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑤)
2		ax-5 1839	. 2 ⊢ (𝜑 → ∀𝑤𝜑)
3		axc11rvOLD 2140	. 2 ⊢ (∀𝑧 𝑧 = 𝑤 → (∀𝑤𝜑 → ∀𝑧𝜑))
4	1, 2, 3	syl2im 40	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))

Syntax hints:   → wi 4  ∀wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)