Theorem vtocl2ga 3557

Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 20-Aug-1995.) Avoid ax-10 2141 and ax-11 2157. (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 340	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ 𝐶 → 𝜓) ↔ (𝐴 ∈ 𝐶 → 𝜒)))
3		vtocl2ga.1	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	3	imbi2d 340	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑦 ∈ 𝐷 → 𝜑) ↔ (𝑦 ∈ 𝐷 → 𝜓)))
5		vtocl2ga.3	. . . . . 6 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜑)
6	5	ex 412	. . . . 5 ⊢ (𝑥 ∈ 𝐶 → (𝑦 ∈ 𝐷 → 𝜑))
7	4, 6	vtoclga 3556	. . . 4 ⊢ (𝐴 ∈ 𝐶 → (𝑦 ∈ 𝐷 → 𝜓))
8	7	com12 32	. . 3 ⊢ (𝑦 ∈ 𝐷 → (𝐴 ∈ 𝐶 → 𝜓))
9	2, 8	vtoclga 3556	. 2 ⊢ (𝐵 ∈ 𝐷 → (𝐴 ∈ 𝐶 → 𝜒))
10	9	impcom 407	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝜒)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809

This theorem is referenced by:  vtocl3ga  3562  vtocl4ga  3565  solin  5588  caovcan  7611  xpord2pred  8144  pwfseqlem2  10673  mulcanenq  10974  ltaddnq  10988  ltrnq  10993  genpv  11013  wrdind  14740  fsumrelem  15823  imasleval  17555  fullfunc  17921  fthfunc  17922  symggrplem  18862  pf1ind  22293  mretopd  23030  dvlip  25950  scvxcvx  26948  issubgoilem  31241  cnre2csqlem  33941