Theorem spv 1858

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 1809	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1351

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 1445  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532

This theorem depends on definitions:  df-bi 117  df-nf 1459

This theorem is referenced by:  spvv  1905  cbvalvw  1917  chvarv  1935  ru  2959  nalset  4128  tfisi  4580  tfr1onlemsucfn  6331  tfr1onlemsucaccv  6332  tfr1onlembxssdm  6334  tfr1onlembfn  6335  tfr1onlemres  6340  tfri1dALT  6342  tfrcllemsucfn  6344  tfrcllemsucaccv  6345  tfrcllembxssdm  6347  tfrcllembfn  6348  tfrcllemres  6353  findcard2  6879  findcard2s  6880  bj-nalset  14205