Theorem vtoclgf 2913

Description: Implicit substitution of a class for a setvar variable, with bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Proof shortened by Mario Carneiro, 10-Oct-2016.)

Hypotheses

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

Assertion

Ref	Expression
vtoclgf	⊢ (A ∈ V → ψ)

Proof of Theorem vtoclgf

Step	Hyp	Ref	Expression
1		elex 2867	. 2 ⊢ (A ∈ V → A ∈ V)
2		vtoclgf.1	. . . 4 ⊢ ℲxA
3	2	issetf 2864	. . 3 ⊢ (A ∈ V ↔ ∃x x = A)
4		vtoclgf.2	. . . 4 ⊢ Ⅎxψ
5		vtoclgf.4	. . . . 5 ⊢ φ
6		vtoclgf.3	. . . . 5 ⊢ (x = A → (φ ↔ ψ))
7	5, 6	mpbii 202	. . . 4 ⊢ (x = A → ψ)
8	4, 7	exlimi 1803	. . 3 ⊢ (∃x x = A → ψ)
9	3, 8	sylbi 187	. 2 ⊢ (A ∈ V → ψ)
10	1, 9	syl 15	1 ⊢ (A ∈ V → ψ)

Syntax hints:   → wi 4   ↔ wb 176  ∃wex 1541  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476  Vcvv 2859

This theorem was proved from 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 theorem 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 2478  df-v 2861

This theorem is referenced by:  vtoclg  2914  vtocl2gf  2916  vtocl3gf  2917  vtoclgaf  2919  ceqsexg  2970  elabgf  2983  mob  3018  opeliunxp2  4822