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Theorem r19.29af2 2446
 Description: A commonly used pattern based on r19.29 2444 (Contributed by Thierry Arnoux, 17-Dec-2017.)
Hypotheses
Ref Expression
r19.29af2.p xφ
r19.29af2.c xχ
r19.29af2.1 (((φ x A) ψ) → χ)
r19.29af2.2 (φx A ψ)
Assertion
Ref Expression
r19.29af2 (φχ)

Proof of Theorem r19.29af2
StepHypRef Expression
1 r19.29af2.2 . . 3 (φx A ψ)
2 r19.29af2.p . . . 4 xφ
3 r19.29af2.1 . . . . 5 (((φ x A) ψ) → χ)
43exp31 346 . . . 4 (φ → (x A → (ψχ)))
52, 4ralrimi 2384 . . 3 (φx A (ψχ))
61, 5jca 290 . 2 (φ → (x A ψ x A (ψχ)))
7 r19.29r 2445 . 2 ((x A ψ x A (ψχ)) → x A (ψ (ψχ)))
8 r19.29af2.c . . 3 xχ
9 pm3.35 329 . . . 4 ((ψ (ψχ)) → χ)
109a1i 9 . . 3 (x A → ((ψ (ψχ)) → χ))
118, 10rexlimi 2420 . 2 (x A (ψ (ψχ)) → χ)
126, 7, 113syl 17 1 (φχ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  Ⅎwnf 1346   ∈ 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-tru 1245  df-nf 1347  df-ral 2305  df-rex 2306 This theorem is referenced by:  r19.29af  2447
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