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Theorem gencl 3266
Description: Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
Hypotheses
Ref Expression
gencl.1 (𝜃 ↔ ∃𝑥(𝜒𝐴 = 𝐵))
gencl.2 (𝐴 = 𝐵 → (𝜑𝜓))
gencl.3 (𝜒𝜑)
Assertion
Ref Expression
gencl (𝜃𝜓)
Distinct variable group:   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜒(𝑥)   𝜃(𝑥)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem gencl
StepHypRef Expression
1 gencl.1 . 2 (𝜃 ↔ ∃𝑥(𝜒𝐴 = 𝐵))
2 gencl.3 . . . . 5 (𝜒𝜑)
3 gencl.2 . . . . 5 (𝐴 = 𝐵 → (𝜑𝜓))
42, 3syl5ib 234 . . . 4 (𝐴 = 𝐵 → (𝜒𝜓))
54impcom 445 . . 3 ((𝜒𝐴 = 𝐵) → 𝜓)
65exlimiv 1898 . 2 (∃𝑥(𝜒𝐴 = 𝐵) → 𝜓)
71, 6sylbi 207 1 (𝜃𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745
This theorem is referenced by:  2gencl  3267  3gencl  3268  indpi  9767  axrrecex  10022
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