Theorem spimvALT 2426

Description: Alternate proof of spimv 2425. Note that it requires only ax-1 6 through ax-5 1943 together with ax6e 2418. Currently, proofs derive from ax6v 2001, but if ax-6 2000 could be used instead, this proof would reduce axiom usage. (Contributed by NM, 31-Jul-1993.) Remove dependency on ax-10 2179. (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 2418	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
2		spimv.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
3	1, 2	eximii 1870	. 2 ⊢ ∃𝑥(𝜑 → 𝜓)
4	3	19.36iv 1979	1 ⊢ (∀𝑥𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2216  ax-13 2407

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by: (None)