Theorem 3impdirp1 42389

Description: A deduction unionizing a non-unionized collection of virtual hypotheses. Commuted version of 3impdir 1349. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
3impdirp1.1	⊢ (((𝜒 ∧ 𝜓) ∧ (𝜑 ∧ 𝜓)) → 𝜃)

Assertion

Ref	Expression
3impdirp1	⊢ ((𝜑 ∧ 𝜒 ∧ 𝜓) → 𝜃)

Proof of Theorem 3impdirp1

Step	Hyp	Ref	Expression
1		ancom 460	. . 3 ⊢ (((𝜒 ∧ 𝜓) ∧ (𝜑 ∧ 𝜓)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜓)))
2		3impdirp1.1	. . 3 ⊢ (((𝜒 ∧ 𝜓) ∧ (𝜑 ∧ 𝜓)) → 𝜃)
3	1, 2	sylbir 234	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜓)) → 𝜃)
4	3	3impdir 1349	1 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜓) → 𝜃)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1087

This theorem is referenced by: (None)