Theorem rexbid 3280

Description: Formula-building rule for restricted existential quantifier (deduction form). For a version based on fewer axioms see rexbidv 3185. (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 3278	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ↔ wb 206  Ⅎwnf 1781   ∈ wcel 2108  ∃wrex 3076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-nf 1782  df-rex 3077

This theorem is referenced by:  rexbidvALT  3281  rexeqbid  3365  scott0  9955  infcvgaux1i  15905  bnj1463  35031  fvineqsneq  37378  poimirlem25  37605  poimirlem26  37606  elrnmptf  45088  smfsupmpt  46736  smfinfmpt  46740