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Theorem cgsexg 2583
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 280 . . 3 ((χ φ) → ψ)
32exlimiv 1486 . 2 (x(χ φ) → ψ)
4 elisset 2562 . . . 4 (A 𝑉x x = A)
5 cgsexg.1 . . . . 5 (x = Aχ)
65eximi 1488 . . . 4 (x x = Axχ)
74, 6syl 14 . . 3 (A 𝑉xχ)
81biimprcd 149 . . . . 5 (ψ → (χφ))
98ancld 308 . . . 4 (ψ → (χ → (χ φ)))
109eximdv 1757 . . 3 (ψ → (xχx(χ φ)))
117, 10syl5com 26 . 2 (A 𝑉 → (ψx(χ φ)))
123, 11impbid2 131 1 (A 𝑉 → (x(χ φ) ↔ ψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553
This theorem is referenced by: (None)
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