Theorem ralrmo3 38963

Description: Pull a restricted universal quantifier into the body (for ∃*). (Contributed by Peter Mazsa, 9-May-2019.)

Assertion

Ref	Expression
ralrmo3	⊢ (∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝜑))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑦)   𝐵(𝑦)

Proof of Theorem ralrmo3

Step	Hyp	Ref	Expression
1		df-ral 3087	. 2 ⊢ (∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐵 → ∃*𝑥 ∈ 𝐴 𝜑))
2		nfv 1942	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
3	2	rmoanim 3856	. . 3 ⊢ (∃*𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝜑) ↔ (𝑦 ∈ 𝐵 → ∃*𝑥 ∈ 𝐴 𝜑))
4	3	albii 1847	. 2 ⊢ (∀𝑦∃*𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐵 → ∃*𝑥 ∈ 𝐴 𝜑))
5	1, 4	bitr4i 281	1 ⊢ (∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝜑))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1566   ∈ wcel 2150  ∀wral 3086  ∃*wrmo 3375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-12 2220

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1808  df-nf 1812  df-mo 2574  df-ral 3087  df-rmo 3376

This theorem is referenced by:  raldmqseu  38964