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Theorem syl31anc 1137
Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
Hypotheses
Ref Expression
sylXanc.1 (φψ)
sylXanc.2 (φχ)
sylXanc.3 (φθ)
sylXanc.4 (φτ)
syl31anc.5 (((ψ χ θ) τ) → η)
Assertion
Ref Expression
syl31anc (φη)

Proof of Theorem syl31anc
StepHypRef Expression
1 sylXanc.1 . . 3 (φψ)
2 sylXanc.2 . . 3 (φχ)
3 sylXanc.3 . . 3 (φθ)
41, 2, 33jca 1083 . 2 (φ → (ψ χ θ))
5 sylXanc.4 . 2 (φτ)
6 syl31anc.5 . 2 (((ψ χ θ) τ) → η)
74, 5, 6syl2anc 391 1 (φη)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101
This theorem depends on definitions:  df-bi 110  df-3an 886
This theorem is referenced by:  syl32anc  1142  stoic4b  1319  enq0tr  6416  ltmul12a  7567  lt2msq1  7592  ledivp1  7610  lemul1ad  7646  lemul2ad  7647  lediv2ad  8379  difelfznle  8723  expubnd  8925
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