Theorem spv 2407

Description: Specialization, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker spvv 1999 if possible. (Contributed by NM, 30-Aug-1993.) (New usage is discouraged.)

Hypothesis

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

Assertion

Ref	Expression
spv	⊢ (∀𝑥𝜑 → 𝜓)

Distinct variable group:   𝜓,𝑥

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

Proof of Theorem spv

Step	Hyp	Ref	Expression
1		spv.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2	1	biimpd 231	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
3	2	spimv 2404	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-12 2172  ax-13 2386

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-nf 1781

This theorem is referenced by:  cbvalvOLD  2416  axc11n-16  36068