Theorem spimvALT 2416

Description: Alternate proof of spimv 2415. Note that it requires only ax-1 6 through ax-5 1924 together with ax6e 2408. Currently, proofs derive from ax6v 1982, but if ax-6 1981 could be used instead, this proof would reduce axiom usage. (Contributed by NM, 31-Jul-1993.) Remove dependency on ax-10 2169. (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 2408	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
2		spimv.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
3	1, 2	eximii 1851	. 2 ⊢ ∃𝑥(𝜑 → 𝜓)
4	3	19.36iv 1960	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-12 2206  ax-13 2397

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1794

This theorem is referenced by: (None)