Theorem bija 381

Description: Combine antecedents into a single biconditional. This inference, reminiscent of ja 186, is reversible: The hypotheses can be deduced from the conclusion alone (see pm5.1im 262 and pm5.21im 374). (Contributed by Wolf Lammen, 13-May-2013.)

Hypotheses

Ref	Expression
bija.1	⊢ (𝜑 → (𝜓 → 𝜒))
bija.2	⊢ (¬ 𝜑 → (¬ 𝜓 → 𝜒))

Assertion

Ref	Expression
bija	⊢ ((𝜑 ↔ 𝜓) → 𝜒)

Proof of Theorem bija

Step	Hyp	Ref	Expression
1		biimpr 219	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
2		bija.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	1, 2	syli 39	. 2 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜒))
4		biimp 214	. . . 4 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
5	4	con3d 152	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜓 → ¬ 𝜑))
6		bija.2	. . 3 ⊢ (¬ 𝜑 → (¬ 𝜓 → 𝜒))
7	5, 6	syli 39	. 2 ⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜓 → 𝜒))
8	3, 7	pm2.61d 179	1 ⊢ ((𝜑 ↔ 𝜓) → 𝜒)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205

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 206

This theorem is referenced by:  impimprbi  825  nanass  1502  equvel  2456  ab0w  4304  epnsym  9297  2lgsoddprm  26469  bj-bibibi  34695  wl-aleq  35621  wl-nfeqfb  35622  rp-fakeimass  41017