Theorem oplem1 1057

Description: A specialized lemma for set theory (ordered pair theorem). (Contributed by NM, 18-Oct-1995.) (Proof shortened by Wolf Lammen, 8-Dec-2012.)

Hypotheses

Ref	Expression
oplem1.1	⊢ (𝜑 → (𝜓 ∨ 𝜒))
oplem1.2	⊢ (𝜑 → (𝜃 ∨ 𝜏))
oplem1.3	⊢ (𝜓 ↔ 𝜃)
oplem1.4	⊢ (𝜒 → (𝜃 ↔ 𝜏))

Assertion

Ref	Expression
oplem1	⊢ (𝜑 → 𝜓)

Proof of Theorem oplem1

Step	Hyp	Ref	Expression
1		oplem1.3	. . . . . . 7 ⊢ (𝜓 ↔ 𝜃)
2	1	notbii 320	. . . . . 6 ⊢ (¬ 𝜓 ↔ ¬ 𝜃)
3		oplem1.1	. . . . . . 7 ⊢ (𝜑 → (𝜓 ∨ 𝜒))
4	3	ord 865	. . . . . 6 ⊢ (𝜑 → (¬ 𝜓 → 𝜒))
5	2, 4	biimtrrid 243	. . . . 5 ⊢ (𝜑 → (¬ 𝜃 → 𝜒))
6		oplem1.2	. . . . . 6 ⊢ (𝜑 → (𝜃 ∨ 𝜏))
7	6	ord 865	. . . . 5 ⊢ (𝜑 → (¬ 𝜃 → 𝜏))
8	5, 7	jcad 512	. . . 4 ⊢ (𝜑 → (¬ 𝜃 → (𝜒 ∧ 𝜏)))
9		oplem1.4	. . . . 5 ⊢ (𝜒 → (𝜃 ↔ 𝜏))
10	9	biimpar 477	. . . 4 ⊢ ((𝜒 ∧ 𝜏) → 𝜃)
11	8, 10	syl6 35	. . 3 ⊢ (𝜑 → (¬ 𝜃 → 𝜃))
12	11	pm2.18d 127	. 2 ⊢ (𝜑 → 𝜃)
13	12, 1	sylibr 234	1 ⊢ (𝜑 → 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848

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 207  df-an 396  df-or 849

This theorem is referenced by:  preq1b  4846