Theorem exbi 1880

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 1866	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  exbii  1881  nfbiit  1884  19.19  2268  eubi  2615  axpr  5403  elirrv  9569  bj-2exbi  37265  bj-3exbi  37266  bj-hbyfrbi  37277  2exbi  45131  rexbidar  45196  onfrALTlem1VD  45639  csbxpgVD  45643  csbrngVD  45645  csbunigVD  45647  e2ebindVD  45661  e2ebindALT  45678