Theorem 2ralsng 4639

Description: Substitution expressed in terms of two quantifications over singletons. (Contributed by AV, 22-Dec-2019.)

Hypotheses

Ref	Expression
ralsng.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
2ralsng.1	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
2ralsng	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝜑 ↔ 𝜒))

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

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝜒(𝑥)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem 2ralsng

Step	Hyp	Ref	Expression
1		ralsng.1	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
2	1	ralbidv 3187	. . 3 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ {𝐵}𝜑 ↔ ∀𝑦 ∈ {𝐵}𝜓))
3	2	ralsng 4636	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝜑 ↔ ∀𝑦 ∈ {𝐵}𝜓))
4		2ralsng.1	. . 3 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
5	4	ralsng 4636	. 2 ⊢ (𝐵 ∈ 𝑊 → (∀𝑦 ∈ {𝐵}𝜓 ↔ 𝜒))
6	3, 5	sylan9bb 517	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝜑 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ∀wral 3078  {csn 4584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1565  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-v 3458  df-sn 4585

This theorem is referenced by:  c0snmgmhm  20513  zrrnghm  20588  mat1ghm  22545  mat1mhm  22546  f1resfz0f1d  35468