Theorem bj-spvv 32723

Description: Version of spv 2260 with a dv condition, which does not require ax-7 1935, ax-12 2047, ax-13 2246. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)

Hypothesis

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

Assertion

Ref	Expression
bj-spvv	⊢ (∀𝑥𝜑 → 𝜓)

Distinct variable groups:   𝑥,𝑦   𝜓,𝑥

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

Proof of Theorem bj-spvv

Step	Hyp	Ref	Expression
1		bj-spvv.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2	1	biimpd 219	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
3	2	bj-spimvv 32721	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 196  ∀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-chvarvv  32726  bj-nalset  32794  bj-ru0  32932