Theorem bj-spimvv 32721

Description: Version of spimv 2257 and spimv1 2115 with a dv condition, which does not require ax-13 2246. UPDATE: this is spimvw 1927. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-spimvv.1	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
bj-spimvv	⊢ (∀𝑥𝜑 → 𝜓)

Distinct variable groups:   𝑥,𝑦   𝜓,𝑥

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

Proof of Theorem bj-spimvv

Step	Hyp	Ref	Expression
1		ax6ev 1890	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
2		bj-spimvv.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
3	1, 2	eximii 1764	. 2 ⊢ ∃𝑥(𝜑 → 𝜓)
4	3	19.36iv 1905	1 ⊢ (∀𝑥𝜑 → 𝜓)

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

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by:  bj-spvv  32723  bj-el  32796