Theorem spimvALT 2425

Description: Alternate proof of spimv 2424. Note that it requires only ax-1 6 through ax-5 1933 together with ax6e 2417. Currently, proofs derive from ax6v 1991, but if ax-6 1990 could be used instead, this proof would reduce axiom usage. (Contributed by NM, 31-Jul-1993.) Remove dependency on ax-10 2178. (Revised by BJ, 29-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
spimv.1	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
spimvALT	⊢ (∀𝑥𝜑 → 𝜓)

Distinct variable group:   𝜓,𝑥

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

Proof of Theorem spimvALT

Step	Hyp	Ref	Expression
1		ax6e 2417	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
2		spimv.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
3	1, 2	eximii 1860	. 2 ⊢ ∃𝑥(𝜑 → 𝜓)
4	3	19.36iv 1969	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-12 2215  ax-13 2406

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803

This theorem is referenced by: (None)