Theorem axc16ALT 2489

Description: Alternate proof of axc16 2264, shorter but requiring ax-10 2144, ax-11 2160, ax-13 2372 and using df-nf 1785 and df-sb 2068. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axc16ALT	⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem axc16ALT

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbequ12 2254	. 2 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
2		ax-5 1911	. . 3 ⊢ (𝜑 → ∀𝑧𝜑)
3	2	hbsb3 2487	. 2 ⊢ ([𝑧 / 𝑥]𝜑 → ∀𝑥[𝑧 / 𝑥]𝜑)
4	1, 3	axc16i 2436	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4  ∀wal 1539  [wsb 2067

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 1968  ax-7 2009  ax-10 2144  ax-11 2160  ax-12 2180  ax-13 2372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1781  df-nf 1785  df-sb 2068

This theorem is referenced by:  axc16gALT  2490