Theorem spv 2325

Description: Specialization, using implicit substitution. See spvv 1956 for a version using fewer axioms. (Contributed by NM, 30-Aug-1993.)

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 221	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
3	2	spimv 2322	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-12 2107  ax-13 2302

This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1744  df-nf 1748

This theorem is referenced by:  cbvalvOLD  2334  isowe2  6924  tfisi  7387  findcard2  8551  marypha1lem  8690  setind  8968  karden  9116  axgroth3  10049  ramcl  16219  alexsubALTlem3  22376  i1fd  24000  dfpo2  32548  dfon2lem6  32590  trer  33222  axc11n-16  35556  elsetrecslem  44202