Theorem vtoclri 2673

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 21-Nov-1994.)

Hypotheses

Ref	Expression
vtoclri.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
vtoclri.2	⊢ ∀𝑥 ∈ 𝐵 𝜑

Assertion

Ref	Expression
vtoclri	⊢ (𝐴 ∈ 𝐵 → 𝜓)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem vtoclri

Step	Hyp	Ref	Expression
1		vtoclri.1	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
2		vtoclri.2	. . 3 ⊢ ∀𝑥 ∈ 𝐵 𝜑
3	2	rspec 2415	. 2 ⊢ (𝑥 ∈ 𝐵 → 𝜑)
4	1, 3	vtoclga 2664	1 ⊢ (𝐴 ∈ 𝐵 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 103   = wceq 1284   ∈ wcel 1433  ∀wral 2348

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603

This theorem is referenced by:  ordpwsucexmid  4313  bj-nn0suc0  10745