Theorem ralbida 3276

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 3266	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒))
5	2	biimprd 248	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜒 → 𝜓))
6	1, 5	ralimdaa 3266	. 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 1781   ∈ wcel 2108  ∀wral 3067

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-ral 3068

This theorem is referenced by:  ralbid  3279  2ralbida  3289  naddsuc2  8757  ac6num  10548  neiptopreu  23162  istrkg2ld  28486  funcnv5mpt  32686  nadd1suc  43354  xrralrecnnge  45305  climf2  45587  clim2f2  45591  limsupub  45625  climinfmpt  45636  limsupubuzmpt  45640  limsupre2mpt  45651  limsupre3mpt  45655  limsupreuzmpt  45660  xlimmnfmpt  45764  xlimpnfmpt  45765  smfsupmpt  46736  smfinfmpt  46740