Theorem pm4.38 634

Description: Theorem *4.38 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)

Assertion

Ref	Expression
pm4.38	⊢ (((𝜑 ↔ 𝜒) ∧ (𝜓 ↔ 𝜃)) → ((𝜑 ∧ 𝜓) ↔ (𝜒 ∧ 𝜃)))

Proof of Theorem pm4.38

Step	Hyp	Ref	Expression
1		simpl 482	. 2 ⊢ (((𝜑 ↔ 𝜒) ∧ (𝜓 ↔ 𝜃)) → (𝜑 ↔ 𝜒))
2		simpr 484	. 2 ⊢ (((𝜑 ↔ 𝜒) ∧ (𝜓 ↔ 𝜃)) → (𝜓 ↔ 𝜃))
3	1, 2	anbi12d 630	1 ⊢ (((𝜑 ↔ 𝜒) ∧ (𝜓 ↔ 𝜃)) → ((𝜑 ∧ 𝜓) ↔ (𝜒 ∧ 𝜃)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395

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

This theorem is referenced by:  bi2anan9  635  xpf1o  8875  isprm3  16316  csbingVD  42393  csbxpgVD  42403  csbunigVD  42407  ichan  44795