Theorem spv 1906

Description: Specialization, using implicit substitition. (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 144	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
3	2	spimv 1857	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-nf 1507

This theorem is referenced by:  spvv  1954  cbvalvw  1966  chvarv  1988  ru  3027  nalset  4214  tfisi  4679  tfr1onlemsucfn  6492  tfr1onlemsucaccv  6493  tfr1onlembxssdm  6495  tfr1onlembfn  6496  tfr1onlemres  6501  tfri1dALT  6503  tfrcllemsucfn  6505  tfrcllemsucaccv  6506  tfrcllembxssdm  6508  tfrcllembfn  6509  tfrcllemres  6514  findcard2  7059  findcard2s  7060  bj-nalset  16313