Theorem ralbida 3249

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 3239	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒))
5	2	biimprd 248	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜒 → 𝜓))
6	1, 5	ralimdaa 3239	. 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 1785   ∈ wcel 2114  ∀wral 3052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-12 2185

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-nf 1786  df-ral 3053

This theorem is referenced by:  ralbid  3251  2ralbida  3261  naddsuc2  8641  ac6num  10403  neiptopreu  23094  istrkg2ld  28549  istrkg2ldOLD  28550  funcnv5mpt  32763  nadd1suc  43778  xrralrecnnge  45777  climf2  46053  clim2f2  46057  limsupub  46091  climinfmpt  46102  limsupubuzmpt  46106  limsupre2mpt  46117  limsupre3mpt  46121  limsupreuzmpt  46126  xlimmnfmpt  46230  xlimpnfmpt  46231  smfsupmpt  47202  smfinfmpt  47206