Theorem vtocldf 3229

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 449	. . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)))
6	3, 5	alrimi 2069	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓 ↔ 𝜒)))
7		vtocld.3	. . 3 ⊢ (𝜑 → 𝜓)
8	3, 7	alrimi 2069	. 2 ⊢ (𝜑 → ∀𝑥𝜓)
9		vtocld.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
10		vtoclgft 3227	. 2 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜒) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) ∧ ∀𝑥𝜓) ∧ 𝐴 ∈ 𝑉) → 𝜒)
11	1, 2, 6, 8, 9, 10	syl221anc 1329	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  Ⅎwnfc 2738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175

This theorem is referenced by:  vtocld  3230  iota2df  5792  riotasv2d  33261