Theorem axc16g 2134

Description: Generalization of axc16 2135. Use the latter when sufficient. This proof only requires, on top of { ax-1 6-- ax-7 1935 }, theorem ax12v 2048. (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.)

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 1981	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑤)
2		ax12v 2048	. . . 4 ⊢ (𝑧 = 𝑤 → (𝜑 → ∀𝑧(𝑧 = 𝑤 → 𝜑)))
3	2	sps 2055	. . 3 ⊢ (∀𝑧 𝑧 = 𝑤 → (𝜑 → ∀𝑧(𝑧 = 𝑤 → 𝜑)))
4		pm2.27 42	. . . 4 ⊢ (𝑧 = 𝑤 → ((𝑧 = 𝑤 → 𝜑) → 𝜑))
5	4	al2imi 1743	. . 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 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:  axc16  2135  axc16gb  2136  axc16nf  2137  axc11v  2138  axc11rv  2139  aevOLD  2162  axc16nfALT  2323