Theorem exbi 1879

Description: Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)

Assertion

Ref	Expression
exbi	⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))

Proof of Theorem exbi

Step	Hyp	Ref	Expression
1		id 23	. 2 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 ↔ 𝜓))
2	1	alexbii 1865	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1567  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841

This proof depends on definitions:  df-bi 210  df-ex 1812

This theorem is used by:  exbii  1880  nfbiit  1883  19.19  2265  eubi  2609  axpr  5388  elirrv  9594  bj-2exbi  37593  bj-3exbi  37594  bj-hbyfrbi  37605  2exbi  45457  rexbidar  45522  onfrALTlem1VD  45965  csbxpgVD  45969  csbrngVD  45971  csbunigVD  45973  e2ebindVD  45987  e2ebindALT  46004