Theorem vtoclri 2805

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 2522	. 2 ⊢ (𝑥 ∈ 𝐵 → 𝜑)
4	1, 3	vtoclga 2796	1 ⊢ (𝐴 ∈ 𝐵 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1348   ∈ wcel 2141  ∀wral 2448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732

This theorem is referenced by:  ordpwsucexmid  4554  bj-nn0suc0  13985