MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  3gencl Structured version   Visualization version   GIF version

Theorem 3gencl 3232
Description: Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
Hypotheses
Ref Expression
3gencl.1 (𝐷𝑆 ↔ ∃𝑥𝑅 𝐴 = 𝐷)
3gencl.2 (𝐹𝑆 ↔ ∃𝑦𝑅 𝐵 = 𝐹)
3gencl.3 (𝐺𝑆 ↔ ∃𝑧𝑅 𝐶 = 𝐺)
3gencl.4 (𝐴 = 𝐷 → (𝜑𝜓))
3gencl.5 (𝐵 = 𝐹 → (𝜓𝜒))
3gencl.6 (𝐶 = 𝐺 → (𝜒𝜃))
3gencl.7 ((𝑥𝑅𝑦𝑅𝑧𝑅) → 𝜑)
Assertion
Ref Expression
3gencl ((𝐷𝑆𝐹𝑆𝐺𝑆) → 𝜃)
Distinct variable groups:   𝑥,𝑦,𝑧   𝑦,𝐷,𝑧   𝑧,𝐹   𝑥,𝑅,𝑦   𝑦,𝑆,𝑧   𝜓,𝑥   𝜒,𝑦   𝜃,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑦,𝑧)   𝜒(𝑥,𝑧)   𝜃(𝑥,𝑦)   𝐴(𝑥,𝑦,𝑧)   𝐵(𝑥,𝑦,𝑧)   𝐶(𝑥,𝑦,𝑧)   𝐷(𝑥)   𝑅(𝑧)   𝑆(𝑥)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦,𝑧)

Proof of Theorem 3gencl
StepHypRef Expression
1 3gencl.3 . . . . 5 (𝐺𝑆 ↔ ∃𝑧𝑅 𝐶 = 𝐺)
2 df-rex 2915 . . . . 5 (∃𝑧𝑅 𝐶 = 𝐺 ↔ ∃𝑧(𝑧𝑅𝐶 = 𝐺))
31, 2bitri 264 . . . 4 (𝐺𝑆 ↔ ∃𝑧(𝑧𝑅𝐶 = 𝐺))
4 3gencl.6 . . . . 5 (𝐶 = 𝐺 → (𝜒𝜃))
54imbi2d 330 . . . 4 (𝐶 = 𝐺 → (((𝐷𝑆𝐹𝑆) → 𝜒) ↔ ((𝐷𝑆𝐹𝑆) → 𝜃)))
6 3gencl.1 . . . . . 6 (𝐷𝑆 ↔ ∃𝑥𝑅 𝐴 = 𝐷)
7 3gencl.2 . . . . . 6 (𝐹𝑆 ↔ ∃𝑦𝑅 𝐵 = 𝐹)
8 3gencl.4 . . . . . . 7 (𝐴 = 𝐷 → (𝜑𝜓))
98imbi2d 330 . . . . . 6 (𝐴 = 𝐷 → ((𝑧𝑅𝜑) ↔ (𝑧𝑅𝜓)))
10 3gencl.5 . . . . . . 7 (𝐵 = 𝐹 → (𝜓𝜒))
1110imbi2d 330 . . . . . 6 (𝐵 = 𝐹 → ((𝑧𝑅𝜓) ↔ (𝑧𝑅𝜒)))
12 3gencl.7 . . . . . . 7 ((𝑥𝑅𝑦𝑅𝑧𝑅) → 𝜑)
13123expia 1265 . . . . . 6 ((𝑥𝑅𝑦𝑅) → (𝑧𝑅𝜑))
146, 7, 9, 11, 132gencl 3231 . . . . 5 ((𝐷𝑆𝐹𝑆) → (𝑧𝑅𝜒))
1514com12 32 . . . 4 (𝑧𝑅 → ((𝐷𝑆𝐹𝑆) → 𝜒))
163, 5, 15gencl 3230 . . 3 (𝐺𝑆 → ((𝐷𝑆𝐹𝑆) → 𝜃))
1716com12 32 . 2 ((𝐷𝑆𝐹𝑆) → (𝐺𝑆𝜃))
18173impia 1259 1 ((𝐷𝑆𝐹𝑆𝐺𝑆) → 𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1481  wex 1702  wcel 1988  wrex 2910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837
This theorem depends on definitions:  df-bi 197  df-an 386  df-3an 1038  df-ex 1703  df-rex 2915
This theorem is referenced by:  axpre-lttrn  9972  axpre-ltadd  9973
  Copyright terms: Public domain W3C validator