Theorem 1fpid3 1023

Description: The value of the conditional operator for propositions is its third argument if the first and second argument imply the third argument. (Contributed by AV, 4-Apr-2021.)

Hypothesis

Ref	Expression
1fpid3.1	⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Assertion

Ref	Expression
1fpid3	⊢ (if-(𝜑, 𝜓, 𝜒) → 𝜒)

Proof of Theorem 1fpid3

Step	Hyp	Ref	Expression
1		df-ifp 1007	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
2		1fpid3.1	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
3		simpr 476	. . 3 ⊢ ((¬ 𝜑 ∧ 𝜒) → 𝜒)
4	2, 3	jaoi 393	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) → 𝜒)
5	1, 4	sylbi 206	1 ⊢ (if-(𝜑, 𝜓, 𝜒) → 𝜒)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383  if-wif 1006

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 196  df-or 384  df-an 385  df-ifp 1007

This theorem is referenced by:  1wlkvtxeledglem  40827