Theorem prlem1 1051

Description: A specialized lemma for set theory (to derive the Axiom of Pairing). (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 5-Jan-2013.)

Hypotheses

Ref	Expression
prlem1.1	⊢ (𝜑 → (𝜂 ↔ 𝜒))
prlem1.2	⊢ (𝜓 → ¬ 𝜃)

Assertion

Ref	Expression
prlem1	⊢ (𝜑 → (𝜓 → (((𝜓 ∧ 𝜒) ∨ (𝜃 ∧ 𝜏)) → 𝜂)))

Proof of Theorem prlem1

Step	Hyp	Ref	Expression
1		prlem1.1	. . . . 5 ⊢ (𝜑 → (𝜂 ↔ 𝜒))
2	1	biimprd 247	. . . 4 ⊢ (𝜑 → (𝜒 → 𝜂))
3	2	adantld 490	. . 3 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜂))
4		prlem1.2	. . . . 5 ⊢ (𝜓 → ¬ 𝜃)
5	4	pm2.21d 121	. . . 4 ⊢ (𝜓 → (𝜃 → 𝜂))
6	5	adantrd 491	. . 3 ⊢ (𝜓 → ((𝜃 ∧ 𝜏) → 𝜂))
7	3, 6	jaao 951	. 2 ⊢ ((𝜑 ∧ 𝜓) → (((𝜓 ∧ 𝜒) ∨ (𝜃 ∧ 𝜏)) → 𝜂))
8	7	ex 412	1 ⊢ (𝜑 → (𝜓 → (((𝜓 ∧ 𝜒) ∨ (𝜃 ∧ 𝜏)) → 𝜂)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843

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-or 844

This theorem is referenced by:  zfpair  5347