Theorem rexbid 3250

Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 27-Jun-1998.)

Hypotheses

Ref	Expression
rexbid.1	⊢ Ⅎ𝑥𝜑
rexbid.2	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

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

Proof of Theorem rexbid

Step	Hyp	Ref	Expression
1		rexbid.1	. 2 ⊢ Ⅎ𝑥𝜑
2		rexbid.2	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	2	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))
4	1, 3	rexbida 3248	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ↔ wb 205  Ⅎwnf 1789   ∈ wcel 2109  ∃wrex 3066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-12 2174

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1786  df-nf 1790  df-rex 3071

This theorem is referenced by:  rexbidvALT  3251  rexeqbid  3351  scott0  9628  infcvgaux1i  15550  bnj1463  33014  fvineqsneq  35562  poimirlem25  35781  poimirlem26  35782  elrnmptf  42671  smfsupmpt  44299  smfinfmpt  44303