Theorem ralbida 3243

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 229	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒))
4	1, 3	ralimdaa 3233	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒))
5	2	biimprd 248	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜒 → 𝜓))
6	1, 5	ralimdaa 3233	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜒 → ∀𝑥 ∈ 𝐴 𝜓))
7	4, 6	impbid 212	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  Ⅎwnf 1784   ∈ wcel 2111  ∀wral 3047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-12 2180

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-nf 1785  df-ral 3048

This theorem is referenced by:  ralbid  3245  2ralbida  3255  naddsuc2  8616  ac6num  10370  neiptopreu  23048  istrkg2ld  28438  funcnv5mpt  32650  nadd1suc  43484  xrralrecnnge  45487  climf2  45763  clim2f2  45767  limsupub  45801  climinfmpt  45812  limsupubuzmpt  45816  limsupre2mpt  45827  limsupre3mpt  45831  limsupreuzmpt  45836  xlimmnfmpt  45940  xlimpnfmpt  45941  smfsupmpt  46912  smfinfmpt  46916