Theorem axc16g 2261

Description: Generalization of axc16 2262. Use the latter when sufficient. This proof only requires, on top of { ax-1 6-- ax-7 2015 }, theorem ax12v 2178. (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 2390, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Revised by BJ, 7-Jul-2021.) Shorten axc11rv 2266. (Revised by Wolf Lammen, 11-Oct-2021.)

Assertion

Ref	Expression
axc16g	⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem axc16g

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		aevlem 2060	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑤)
2		ax12v 2178	. . . 4 ⊢ (𝑧 = 𝑤 → (𝜑 → ∀𝑧(𝑧 = 𝑤 → 𝜑)))
3	2	sps 2184	. . 3 ⊢ (∀𝑧 𝑧 = 𝑤 → (𝜑 → ∀𝑧(𝑧 = 𝑤 → 𝜑)))
4		pm2.27 42	. . . 4 ⊢ (𝑧 = 𝑤 → ((𝑧 = 𝑤 → 𝜑) → 𝜑))
5	4	al2imi 1816	. . 3 ⊢ (∀𝑧 𝑧 = 𝑤 → (∀𝑧(𝑧 = 𝑤 → 𝜑) → ∀𝑧𝜑))
6	3, 5	syld 47	. 2 ⊢ (∀𝑧 𝑧 = 𝑤 → (𝜑 → ∀𝑧𝜑))
7	1, 6	syl 17	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))

Syntax hints:   → wi 4  ∀wal 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781

This theorem is referenced by:  axc16  2262  axc16gb  2263  axc16nf  2264  axc11v  2265  axc16nfALT  2459