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Theorem 2gencl 2604
Description: Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
Hypotheses
Ref Expression
2gencl.1 (𝐶𝑆 ↔ ∃𝑥𝑅 𝐴 = 𝐶)
2gencl.2 (𝐷𝑆 ↔ ∃𝑦𝑅 𝐵 = 𝐷)
2gencl.3 (𝐴 = 𝐶 → (𝜑𝜓))
2gencl.4 (𝐵 = 𝐷 → (𝜓𝜒))
2gencl.5 ((𝑥𝑅𝑦𝑅) → 𝜑)
Assertion
Ref Expression
2gencl ((𝐶𝑆𝐷𝑆) → 𝜒)
Distinct variable groups:   𝑥,𝑦   𝑥,𝑅   𝜓,𝑥   𝑦,𝐶   𝑦,𝑆   𝜒,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝜒(𝑥)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥)   𝐷(𝑥,𝑦)   𝑅(𝑦)   𝑆(𝑥)

Proof of Theorem 2gencl
StepHypRef Expression
1 2gencl.2 . . . 4 (𝐷𝑆 ↔ ∃𝑦𝑅 𝐵 = 𝐷)
2 df-rex 2329 . . . 4 (∃𝑦𝑅 𝐵 = 𝐷 ↔ ∃𝑦(𝑦𝑅𝐵 = 𝐷))
31, 2bitri 177 . . 3 (𝐷𝑆 ↔ ∃𝑦(𝑦𝑅𝐵 = 𝐷))
4 2gencl.4 . . . 4 (𝐵 = 𝐷 → (𝜓𝜒))
54imbi2d 223 . . 3 (𝐵 = 𝐷 → ((𝐶𝑆𝜓) ↔ (𝐶𝑆𝜒)))
6 2gencl.1 . . . . . 6 (𝐶𝑆 ↔ ∃𝑥𝑅 𝐴 = 𝐶)
7 df-rex 2329 . . . . . 6 (∃𝑥𝑅 𝐴 = 𝐶 ↔ ∃𝑥(𝑥𝑅𝐴 = 𝐶))
86, 7bitri 177 . . . . 5 (𝐶𝑆 ↔ ∃𝑥(𝑥𝑅𝐴 = 𝐶))
9 2gencl.3 . . . . . 6 (𝐴 = 𝐶 → (𝜑𝜓))
109imbi2d 223 . . . . 5 (𝐴 = 𝐶 → ((𝑦𝑅𝜑) ↔ (𝑦𝑅𝜓)))
11 2gencl.5 . . . . . 6 ((𝑥𝑅𝑦𝑅) → 𝜑)
1211ex 112 . . . . 5 (𝑥𝑅 → (𝑦𝑅𝜑))
138, 10, 12gencl 2603 . . . 4 (𝐶𝑆 → (𝑦𝑅𝜓))
1413com12 30 . . 3 (𝑦𝑅 → (𝐶𝑆𝜓))
153, 5, 14gencl 2603 . 2 (𝐷𝑆 → (𝐶𝑆𝜒))
1615impcom 120 1 ((𝐶𝑆𝐷𝑆) → 𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wex 1397  wcel 1409  wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-gen 1354  ax-ie2 1399  ax-17 1435
This theorem depends on definitions:  df-bi 114  df-rex 2329
This theorem is referenced by:  3gencl  2605  axaddrcl  6998  axmulrcl  7000  axpre-apti  7016  axpre-mulgt0  7018  uzin2  9813
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