Theorem oplem1 930

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 287	. . . . . 6 ⊢ (¬ ψ ↔ ¬ θ)
3		oplem1.1	. . . . . . 7 ⊢ (φ → (ψ ∨ χ))
4	3	ord 366	. . . . . 6 ⊢ (φ → (¬ ψ → χ))
5	2, 4	syl5bir 209	. . . . 5 ⊢ (φ → (¬ θ → χ))
6		oplem1.2	. . . . . 6 ⊢ (φ → (θ ∨ τ))
7	6	ord 366	. . . . 5 ⊢ (φ → (¬ θ → τ))
8	5, 7	jcad 519	. . . 4 ⊢ (φ → (¬ θ → (χ ∧ τ)))
9		oplem1.4	. . . . 5 ⊢ (χ → (θ ↔ τ))
10	9	biimpar 471	. . . 4 ⊢ ((χ ∧ τ) → θ)
11	8, 10	syl6 29	. . 3 ⊢ (φ → (¬ θ → θ))
12	11	pm2.18d 103	. 2 ⊢ (φ → θ)
13	12, 1	sylibr 203	1 ⊢ (φ → ψ)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360

This theorem is used by:  preqr1  4125