Theorem ralbida 3268

Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Oct-2003.) (Proof shortened by Wolf Lammen, 31-Oct-2024.)

Hypotheses

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

Assertion

Ref	Expression
ralbida	⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒))

Proof of Theorem ralbida

Step	Hyp	Ref	Expression
1		ralbida.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		ralbida.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))
3	2	biimpd 228	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒))
4	1, 3	ralimdaa 3258	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒))
5	2	biimprd 247	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜒 → 𝜓))
6	1, 5	ralimdaa 3258	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜒 → ∀𝑥 ∈ 𝐴 𝜓))
7	4, 6	impbid 211	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397  Ⅎwnf 1786   ∈ wcel 2107  ∀wral 3062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-12 2172

This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-nf 1787  df-ral 3063

This theorem is referenced by:  ralbid  3271  2ralbida  3281  ac6num  10474  neiptopreu  22637  istrkg2ld  27711  funcnv5mpt  31893  nadd1suc  42142  naddsuc2  42143  xrralrecnnge  44100  climf2  44382  clim2f2  44386  limsupub  44420  climinfmpt  44431  limsupubuzmpt  44435  limsupre2mpt  44446  limsupre3mpt  44450  limsupreuzmpt  44455  xlimmnfmpt  44559  xlimpnfmpt  44560  smfsupmpt  45531  smfinfmpt  45535