Theorem vtocldf 2781

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 114	. . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)))
6	3, 5	alrimi 1515	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓 ↔ 𝜒)))
7		vtocld.3	. . 3 ⊢ (𝜑 → 𝜓)
8	3, 7	alrimi 1515	. 2 ⊢ (𝜑 → ∀𝑥𝜓)
9		vtocld.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
10		vtoclgft 2780	. 2 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜒) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) ∧ ∀𝑥𝜓) ∧ 𝐴 ∈ 𝑉) → 𝜒)
11	1, 2, 6, 8, 9, 10	syl221anc 1244	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1346   = wceq 1348  Ⅎwnf 1453   ∈ wcel 2141  Ⅎwnfc 2299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732

This theorem is referenced by:  vtocld  2782  peano2  4579  iota2df  5184