Theorem vtocldf 3555

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 415	. . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)))
6	3, 5	alrimi 2209	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓 ↔ 𝜒)))
7		vtocld.3	. . 3 ⊢ (𝜑 → 𝜓)
8	3, 7	alrimi 2209	. 2 ⊢ (𝜑 → ∀𝑥𝜓)
9		vtocld.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
10		vtoclgft 3553	. 2 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜒) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) ∧ ∀𝑥𝜓) ∧ 𝐴 ∈ 𝑉) → 𝜒)
11	1, 2, 6, 8, 9, 10	syl221anc 1377	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1531   = wceq 1533  Ⅎwnf 1780   ∈ wcel 2110  Ⅎwnfc 2961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-ex 1777  df-nf 1781  df-cleq 2814  df-clel 2893  df-nfc 2963

This theorem is referenced by:  vtocld  3556  iota2df  6341  riotasv2d  36092