Theorem prlem1 928

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 214	. . . 4 ⊢ (φ → (χ → η))
3	2	adantld 453	. . 3 ⊢ (φ → ((ψ ∧ χ) → η))
4		prlem1.2	. . . . 5 ⊢ (ψ → ¬ θ)
5	4	pm2.21d 98	. . . 4 ⊢ (ψ → (θ → η))
6	5	adantrd 454	. . 3 ⊢ (ψ → ((θ ∧ τ) → η))
7	3, 6	jaao 495	. 2 ⊢ ((φ ∧ ψ) → (((ψ ∧ χ) ∨ (θ ∧ τ)) → η))
8	7	ex 423	1 ⊢ (φ → (ψ → (((ψ ∧ χ) ∨ (θ ∧ τ)) → η)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358

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 177  df-or 359  df-an 360

This theorem is referenced by: (None)