Theorem rexbid 3280

Description: Formula-building rule for restricted existential quantifier (deduction form). For a version based on fewer axioms see rexbidv 3190. (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 486	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))
4	1, 3	rexbida 3278	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  Ⅎwnf 1815   ∈ wcel 2146  ∃wrex 3090

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-12 2215

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-nf 1816  df-rex 3091

This theorem is used by:  rexbidvALT  3281  rexeqbid  3348  scott0b  9950  scott0OLD  9951  infcvgaux1i  16041  bnj1463  35741  srepdflem2  36043  fvineqsneq  38478  poimirlem25  38706  poimirlem26  38707  elrnmptf  46338  smfsupmpt  47967  smfinfmpt  47971