Theorem vtocldf 2907

Description: Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
vtocld.1	⊢ (φ → A ∈ V)
vtocld.2	⊢ ((φ ∧ x = A) → (ψ ↔ χ))
vtocld.3	⊢ (φ → ψ)
vtocldf.4	⊢ Ⅎxφ
vtocldf.5	⊢ (φ → ℲxA)
vtocldf.6	⊢ (φ → Ⅎxχ)

Assertion

Ref	Expression
vtocldf	⊢ (φ → χ)

Proof of Theorem vtocldf

Step	Hyp	Ref	Expression
1		vtocldf.5	. 2 ⊢ (φ → ℲxA)
2		vtocldf.6	. 2 ⊢ (φ → Ⅎxχ)
3		vtocldf.4	. . 3 ⊢ Ⅎxφ
4		vtocld.2	. . . 4 ⊢ ((φ ∧ x = A) → (ψ ↔ χ))
5	4	ex 423	. . 3 ⊢ (φ → (x = A → (ψ ↔ χ)))
6	3, 5	alrimi 1765	. 2 ⊢ (φ → ∀x(x = A → (ψ ↔ χ)))
7		vtocld.3	. . 3 ⊢ (φ → ψ)
8	3, 7	alrimi 1765	. 2 ⊢ (φ → ∀xψ)
9		vtocld.1	. 2 ⊢ (φ → A ∈ V)
10		vtoclgft 2906	. 2 ⊢ (((ℲxA ∧ Ⅎxχ) ∧ (∀x(x = A → (ψ ↔ χ)) ∧ ∀xψ) ∧ A ∈ V) → χ)
11	1, 2, 6, 8, 9, 10	syl221anc 1193	1 ⊢ (φ → χ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2477

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-an 360  df-3an 936  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 referenced by:  vtocld  2908