Theorem r19.12sn 4670

Description: Special case of r19.12 3281 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 4621	. 2 ⊢ (∃𝑥 ∈ {𝐴}∀𝑦 ∈ 𝐵 𝜑 ↔ [𝐴 / 𝑥]∀𝑦 ∈ 𝐵 𝜑)
3		rexsns 4621	. . 3 ⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑)
4	3	ralbii 3078	. 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 2111  ∀wral 3047  ∃wrex 3056  [wsbc 3736  {csn 4573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-v 3438  df-sbc 3737  df-sn 4574

This theorem is referenced by:  intimasn  43749