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Theorem r19.29vva 2450
 Description: A commonly used pattern based on r19.29 2444, version with two restricted quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.)
Hypotheses
Ref Expression
r19.29vva.1 ((((φ x A) y B) ψ) → χ)
r19.29vva.2 (φx A y B ψ)
Assertion
Ref Expression
r19.29vva (φχ)
Distinct variable groups:   y,A   x,y,χ   φ,x,y
Allowed substitution hints:   ψ(x,y)   A(x)   B(x,y)

Proof of Theorem r19.29vva
StepHypRef Expression
1 r19.29vva.1 . . . . . 6 ((((φ x A) y B) ψ) → χ)
21ex 108 . . . . 5 (((φ x A) y B) → (ψχ))
32ralrimiva 2386 . . . 4 ((φ x A) → y B (ψχ))
43ralrimiva 2386 . . 3 (φx A y B (ψχ))
5 r19.29vva.2 . . 3 (φx A y B ψ)
64, 5r19.29d2r 2449 . 2 (φx A y B ((ψχ) ψ))
7 pm3.35 329 . . . . 5 ((ψ (ψχ)) → χ)
87ancoms 255 . . . 4 (((ψχ) ψ) → χ)
98rexlimivw 2423 . . 3 (y B ((ψχ) ψ) → χ)
109rexlimivw 2423 . 2 (x A y B ((ψχ) ψ) → χ)
116, 10syl 14 1 (φχ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1390  ∀wral 2300  ∃wrex 2301 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-4 1397  ax-17 1416  ax-ial 1424  ax-i5r 1425 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-ral 2305  df-rex 2306 This theorem is referenced by: (None)
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