Theorem vtoclgaf 2920

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 17-Feb-2006.) (Revised by Mario Carneiro, 10-Oct-2016.)

Hypotheses

Ref	Expression
vtoclgaf.1	⊢ ℲxA
vtoclgaf.2	⊢ Ⅎxψ
vtoclgaf.3	⊢ (x = A → (φ ↔ ψ))
vtoclgaf.4	⊢ (x ∈ B → φ)

Assertion

Ref	Expression
vtoclgaf	⊢ (A ∈ B → ψ)

Distinct variable group:   x,B

Allowed substitution hints:   φ(x)   ψ(x)   A(x)

Proof of Theorem vtoclgaf

Step	Hyp	Ref	Expression
1		vtoclgaf.1	. . 3 ⊢ ℲxA
2	1	nfel1 2500	. . . 4 ⊢ Ⅎx A ∈ B
3		vtoclgaf.2	. . . 4 ⊢ Ⅎxψ
4	2, 3	nfim 1813	. . 3 ⊢ Ⅎx(A ∈ B → ψ)
5		eleq1 2413	. . . 4 ⊢ (x = A → (x ∈ B ↔ A ∈ B))
6		vtoclgaf.3	. . . 4 ⊢ (x = A → (φ ↔ ψ))
7	5, 6	imbi12d 311	. . 3 ⊢ (x = A → ((x ∈ B → φ) ↔ (A ∈ B → ψ)))
8		vtoclgaf.4	. . 3 ⊢ (x ∈ B → φ)
9	1, 4, 7, 8	vtoclgf 2914	. 2 ⊢ (A ∈ B → (A ∈ B → ψ))
10	9	pm2.43i 43	1 ⊢ (A ∈ B → ψ)

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2477

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862

This theorem is used by:  vtoclga  2921  ssiun2s  4011  fvmptss  5706  fvmptf  5723