Theorem ee11an 42310

Description: e11an 42309 without virtual deductions. syl22anc 836 is also e11an 42309 without virtual deductions, exept with a different order of hypotheses. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ee11an.1	⊢ (𝜑 → 𝜓)
ee11an.2	⊢ (𝜑 → 𝜒)
ee11an.3	⊢ ((𝜓 ∧ 𝜒) → 𝜃)

Assertion

Ref	Expression
ee11an	⊢ (𝜑 → 𝜃)

Proof of Theorem ee11an

Step	Hyp	Ref	Expression
1		ee11an.1	. 2 ⊢ (𝜑 → 𝜓)
2		ee11an.2	. 2 ⊢ (𝜑 → 𝜒)
3		ee11an.3	. . 3 ⊢ ((𝜓 ∧ 𝜒) → 𝜃)
4	3	ex 413	. 2 ⊢ (𝜓 → (𝜒 → 𝜃))
5	1, 2, 4	sylc 65	1 ⊢ (𝜑 → 𝜃)

Syntax hints:   → wi 4   ∧ wa 396

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 397

This theorem is referenced by: (None)