Theorem oplem1 970

Description: A specialized lemma for set theory (ordered pair theorem). (Contributed by NM, 18-Oct-1995.) (Proof shortened by Wolf Lammen, 8-Dec-2012.) (Proof shortened by Mario Carneiro, 2-Feb-2015.)

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		oplem1.1	. 2 ⊢ (𝜑 → (𝜓 ∨ 𝜒))
2		idd 21	. . 3 ⊢ (𝜑 → (𝜓 → 𝜓))
3		oplem1.2	. . . . 5 ⊢ (𝜑 → (𝜃 ∨ 𝜏))
4		ax-1 6	. . . . . 6 ⊢ (𝜃 → (𝜒 → 𝜃))
5		oplem1.4	. . . . . . 7 ⊢ (𝜒 → (𝜃 ↔ 𝜏))
6	5	biimprcd 159	. . . . . 6 ⊢ (𝜏 → (𝜒 → 𝜃))
7	4, 6	jaoi 711	. . . . 5 ⊢ ((𝜃 ∨ 𝜏) → (𝜒 → 𝜃))
8	3, 7	syl 14	. . . 4 ⊢ (𝜑 → (𝜒 → 𝜃))
9		oplem1.3	. . . 4 ⊢ (𝜓 ↔ 𝜃)
10	8, 9	syl6ibr 161	. . 3 ⊢ (𝜑 → (𝜒 → 𝜓))
11	2, 10	jaod 712	. 2 ⊢ (𝜑 → ((𝜓 ∨ 𝜒) → 𝜓))
12	1, 11	mpd 13	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 104   ∨ wo 703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  preqr1g  3753  preqr1  3755