Theorem rexbida 3047

Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 6-Oct-2003.)

Hypotheses

Ref	Expression
rexbida.1	⊢ Ⅎ𝑥𝜑
rexbida.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
rexbida	⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒))

Proof of Theorem rexbida

Step	Hyp	Ref	Expression
1		rexbida.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		rexbida.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))
3	2	pm5.32da 673	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)))
4	1, 3	exbid 2091	. 2 ⊢ (𝜑 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜒)))
5		df-rex 2918	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
6		df-rex 2918	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜒 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜒))
7	4, 5, 6	3bitr4g 303	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∃wex 1704  Ⅎwnf 1708   ∈ wcel 1990  ∃wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710  df-rex 2918

This theorem is referenced by:  rexbidvaALT  3050  rexbid  3051  dfiun2g  4552  fun11iun  7126  iuneq12daf  29373  bnj1366  30900  glbconxN  34664  supminfrnmpt  39672  limsupre2mpt  39962  limsupre3mpt  39966  limsupreuzmpt  39971