Theorem vtocl2 3531

Description: Implicit substitution of classes for setvar variables. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Hypotheses

Ref	Expression
vtocl2.1	⊢ 𝐴 ∈ V
vtocl2.2	⊢ 𝐵 ∈ V
vtocl2.3	⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
vtocl2.4	⊢ 𝜑

Assertion

Ref	Expression
vtocl2	⊢ 𝜓

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

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝐴(𝑦)

Proof of Theorem vtocl2

Step	Hyp	Ref	Expression
1		vtocl2.2	. 2 ⊢ 𝐵 ∈ V
2		vtocl2.1	. . 3 ⊢ 𝐴 ∈ V
3		vtocl2.3	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
4	3	pm5.74da 815	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑦 = 𝐵 → 𝜑) ↔ (𝑦 = 𝐵 → 𝜓)))
5		vtocl2.4	. . . 4 ⊢ 𝜑
6	5	a1i 11	. . 3 ⊢ (𝑦 = 𝐵 → 𝜑)
7	2, 4, 6	vtocl 3525	. 2 ⊢ (𝑦 = 𝐵 → 𝜓)
8	1, 7	vtocle 3523	1 ⊢ 𝜓

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-clel 2838

This theorem is used by:  vtocl3  3532  caovord  7621  sornom  10265  wloglei  11750  ipodrsima  18601  mpfind  22275  mclsppslem  36083  mh-inf3f1  37080  monotoddzzfi  43697