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Theorem cgsexg 2890
 Description: Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007.)
Hypotheses
Ref Expression
cgsexg.1 (x = Aχ)
cgsexg.2 (χ → (φψ))
Assertion
Ref Expression
cgsexg (A V → (x(χ φ) ↔ ψ))
Distinct variable groups:   x,A   ψ,x
Allowed substitution hints:   φ(x)   χ(x)   V(x)

Proof of Theorem cgsexg
StepHypRef Expression
1 cgsexg.2 . . . 4 (χ → (φψ))
21biimpa 470 . . 3 ((χ φ) → ψ)
32exlimiv 1634 . 2 (x(χ φ) → ψ)
4 elisset 2869 . . . 4 (A Vx x = A)
5 cgsexg.1 . . . . 5 (x = Aχ)
65eximi 1576 . . . 4 (x x = Axχ)
74, 6syl 15 . . 3 (A Vxχ)
81biimprcd 216 . . . . 5 (ψ → (χφ))
98ancld 536 . . . 4 (ψ → (χ → (χ φ)))
109eximdv 1622 . . 3 (ψ → (xχx(χ φ)))
117, 10syl5com 26 . 2 (A V → (ψx(χ φ)))
123, 11impbid2 195 1 (A V → (x(χ φ) ↔ ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861 This theorem is referenced by: (None)
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