Theorem r19.12sn 4676

Description: Special case of r19.12 3284 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 3823	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
2		rexsns 4627	. 2 ⊢ (∃𝑥 ∈ {𝐴}∀𝑦 ∈ 𝐵 𝜑 ↔ [𝐴 / 𝑥]∀𝑦 ∈ 𝐵 𝜑)
3		rexsns 4627	. . 3 ⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑)
4	3	ralbii 3081	. 2 ⊢ (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ {𝐴}𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)
5	1, 2, 4	3bitr4g 314	1 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ {𝐴}𝜑))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2114  ∀wral 3050  ∃wrex 3059  [wsbc 3739  {csn 4579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ral 3051  df-rex 3060  df-v 3441  df-sbc 3740  df-sn 4580

This theorem is referenced by:  intimasn  43935