Theorem vtocl2g 1386

Description: Implicit substitution of 2 classes for 2 set variables.

Hypotheses

Ref	Expression
vtocl2g.1	⊢ (x = A → (φ ↔ ψ))
vtocl2g.2	⊢ (y = B → (ψ ↔ χ))
vtocl2g.3	⊢ φ

Assertion

Ref	Expression
vtocl2g	⊢ ((A ∈ C ∧ B ∈ D) → χ)

Distinct variable group(s):   x,A   y,A   y,B   ψ,x   χ,y

Proof of Theorem vtocl2g

Step	Hyp	Ref	Expression
1		ax-17 925	. 2 ⊢ (ψ → ∀xψ)
2		ax-17 925	. 2 ⊢ (χ → ∀yχ)
3		vtocl2g.1	. 2 ⊢ (x = A → (φ ↔ ψ))
4		vtocl2g.2	. 2 ⊢ (y = B → (ψ ↔ χ))
5		vtocl2g.3	. 2 ⊢ φ
6	1, 2, 3, 4, 5	vtocl2gf 1385	1 ⊢ ((A ∈ C ∧ B ∈ D) → χ)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092

This theorem is referenced by:  unexb 1950  vtoclr 2449  xpexg 2489  opelcog 2511  funbrfv 2852  ensymg 3316  xpsneng 3340  sbth 3359  en2lp 3453  unxpdom 3650  prcdpq 3891

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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