Theorem bnj1219 35430

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1219.1	⊢ (𝜒 ↔ (𝜑 ∧ 𝜓 ∧ 𝜁))
bnj1219.2	⊢ (𝜃 ↔ (𝜒 ∧ 𝜏 ∧ 𝜂))

Assertion

Ref	Expression
bnj1219	⊢ (𝜃 → 𝜓)

Proof of Theorem bnj1219

Step	Hyp	Ref	Expression
1		bnj1219.2	. 2 ⊢ (𝜃 ↔ (𝜒 ∧ 𝜏 ∧ 𝜂))
2		bnj1219.1	. . 3 ⊢ (𝜒 ↔ (𝜑 ∧ 𝜓 ∧ 𝜁))
3	2	simp2bi 1164	. 2 ⊢ (𝜒 → 𝜓)
4	1, 3	bnj835 35390	1 ⊢ (𝜃 → 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105

This theorem is used by:  bnj1379  35460