Theorem r19.12sn 4680

Description: Special case of r19.12 3311 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.) (Revised by BJ, 18-Mar-2020.)

Assertion

Ref	Expression
r19.12sn	⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ {𝐴}𝜑))

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

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

Proof of Theorem r19.12sn

Step	Hyp	Ref	Expression
1		sbcralg 3820	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
2		rexsns 4631	. 2 ⊢ (∃𝑥 ∈ {𝐴}∀𝑦 ∈ 𝐵 𝜑 ↔ [𝐴 / 𝑥]∀𝑦 ∈ 𝐵 𝜑)
3		rexsns 4631	. . 3 ⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑)
4	3	ralbii 3108	. 2 ⊢ (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ {𝐴}𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)
5	1, 2, 4	3bitr4g 317	1 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ {𝐴}𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086  [wsbc 3738  {csn 4583

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-v 3452  df-sbc 3739  df-sn 4584

This theorem is used by:  intimasn  44601