Theorem ralbida 3196

Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Oct-2003.)

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	pm5.74da 800	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐴 → 𝜒)))
4	1, 3	albid 2191	. 2 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜒)))
5		df-ral 3112	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
6		df-ral 3112	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜒))
7	4, 5, 6	3bitr4g 315	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1523  Ⅎwnf 1769   ∈ wcel 2083  ∀wral 3107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-12 2143

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1766  df-nf 1770  df-ral 3112

This theorem is referenced by:  ralbid  3197  2ralbida  3198  ralbiOLD  3199  ac6num  9754  neiptopreu  21429  istrkg2ld  25932  funcnv5mpt  30099  xrralrecnnge  41224  climf2  41510  clim2f2  41514  limsupub  41548  climinfmpt  41559  limsupubuzmpt  41563  limsupre2mpt  41574  limsupre3mpt  41578  limsupreuzmpt  41583  xlimmnfmpt  41687  xlimpnfmpt  41688  smfsupmpt  42653  smfinfmpt  42657