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Theorem r19.29_2a 2434
 Description: A commonly used pattern based on r19.29 2428, version with two restricted quantifiers (Contributed by Thierry Arnoux, 26-Nov-2017.)
Hypotheses
Ref Expression
r19.29_2a.1 ((((φ x A) y B) ψ) → χ)
r19.29_2a.2 (φx A y B ψ)
Assertion
Ref Expression
r19.29_2a (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 r19.29_2a
StepHypRef Expression
1 r19.29_2a.1 . . . . . 6 ((((φ x A) y B) ψ) → χ)
21ex 108 . . . . 5 (((φ x A) y B) → (ψχ))
32ralrimiva 2370 . . . 4 ((φ x A) → y B (ψχ))
43ralrimiva 2370 . . 3 (φx A y B (ψχ))
5 r19.29_2a.2 . . 3 (φx A y B ψ)
64, 5r19.29d2r 2433 . 2 (φx A y B ((ψχ) ψ))
7 pm3.35 329 . . . . 5 ((ψ (ψχ)) → χ)
87ancoms 255 . . . 4 (((ψχ) ψ) → χ)
98rexlimivw 2407 . . 3 (y B ((ψχ) ψ) → χ)
109rexlimivw 2407 . 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 1374  ∀wral 2284  ∃wrex 2285 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 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-17 1400  ax-ial 1409  ax-i5r 1410 This theorem depends on definitions:  df-bi 110  df-nf 1330  df-ral 2289  df-rex 2290 This theorem is referenced by: (None)
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