Theorem oplem1 578

Description: A specialized lemma for set theory (ordered pair theorem).

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	. . . . 5 ⊢ (φ → (ψ ∨ χ))
2	1	ord 202	. . . 4 ⊢ (φ → (¬ ψ → χ))
3		oplem1.2	. . . . . 6 ⊢ (φ → (θ ∨ τ))
4	3	ord 202	. . . . 5 ⊢ (φ → (¬ θ → τ))
5		oplem1.3	. . . . . 6 ⊢ (ψ ↔ θ)
6	5	negbii 162	. . . . 5 ⊢ (¬ ψ ↔ ¬ θ)
7	4, 6	syl5ib 181	. . . 4 ⊢ (φ → (¬ ψ → τ))
8	2, 7	jcad 455	. . 3 ⊢ (φ → (¬ ψ → (χ ∧ τ)))
9		oplem1.4	. . . . 5 ⊢ (χ → (θ ↔ τ))
10	9, 5	syl5bb 410	. . . 4 ⊢ (χ → (ψ ↔ τ))
11	10	biimpar 325	. . 3 ⊢ ((χ ∧ τ) → ψ)
12	8, 11	syl6 23	. 2 ⊢ (φ → (¬ ψ → ψ))
13		pm2.18 75	. 2 ⊢ ((¬ ψ → ψ) → ψ)
14	12, 13	syl 12	1 ⊢ (φ → ψ)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196

This theorem is referenced by:  preqr1 1872

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198

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