Theorem vtocl2ga 3540

Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 20-Aug-1995.) Avoid ax-10 2178 and ax-11 2194. (Revised by GG, 20-Aug-2023.) (Proof shortened by Wolf Lammen, 23-Aug-2023.)

Hypotheses

Ref	Expression
vtocl2ga.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
vtocl2ga.2	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
vtocl2ga.3	⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜑)

Assertion

Ref	Expression
vtocl2ga	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝜒)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝜓,𝑥   𝜒,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝜓(𝑦)   𝜒(𝑥)   𝐵(𝑥)

Proof of Theorem vtocl2ga

Step	Hyp	Ref	Expression
1		vtocl2ga.2	. . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
2	1	imbi2d 343	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ 𝐶 → 𝜓) ↔ (𝐴 ∈ 𝐶 → 𝜒)))
3		vtocl2ga.1	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	3	imbi2d 343	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑦 ∈ 𝐷 → 𝜑) ↔ (𝑦 ∈ 𝐷 → 𝜓)))
5		vtocl2ga.3	. . . . . 6 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜑)
6	5	ex 418	. . . . 5 ⊢ (𝑥 ∈ 𝐶 → (𝑦 ∈ 𝐷 → 𝜑))
7	4, 6	vtoclga 3539	. . . 4 ⊢ (𝐴 ∈ 𝐶 → (𝑦 ∈ 𝐷 → 𝜓))
8	7	com12 33	. . 3 ⊢ (𝑦 ∈ 𝐷 → (𝐴 ∈ 𝐶 → 𝜓))
9	2, 8	vtoclga 3539	. 2 ⊢ (𝐵 ∈ 𝐷 → (𝐴 ∈ 𝐶 → 𝜒))
10	9	impcom 413	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝜒)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837

This theorem is used by:  vtocl3ga  3543  vtocl4ga  3545  solin  5594  caovcan  7621  xpord2pred  8146  pwfseqlem2  10671  mulcanenq  10972  ltaddnq  10986  ltrnq  10991  genpv  11011  wrdind  14793  fsumrelem  15896  imasleval  17631  fullfunc  18001  fthfunc  18002  symggrplem  18994  pf1ind  22581  mretopd  23318  dvlip  26222  scvxcvx  27220  issubgoilem  31727  cnre2csqlem  34407