Theorem wl-dfrexsb 34853

Description: An alternate definition of restricted existential quantification (df-wl-rex 34848) using substitution. (Contributed by Wolf Lammen, 25-May-2023.)

Assertion

Ref	Expression
wl-dfrexsb	⊢ (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑))

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

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem wl-dfrexsb

Step	Hyp	Ref	Expression
1		df-wl-rex 34848	. . 3 ⊢ (∃(𝑥 : 𝐴)𝜑 ↔ ¬ ∀(𝑥 : 𝐴) ¬ 𝜑)
2		wl-dfralsb 34839	. . 3 ⊢ (∀(𝑥 : 𝐴) ¬ 𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐴 → [𝑦 / 𝑥] ¬ 𝜑))
3	1, 2	xchbinx 336	. 2 ⊢ (∃(𝑥 : 𝐴)𝜑 ↔ ¬ ∀𝑦(𝑦 ∈ 𝐴 → [𝑦 / 𝑥] ¬ 𝜑))
4		sbn 2287	. . . . . 6 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
5	4	imbi2i 338	. . . . 5 ⊢ ((𝑦 ∈ 𝐴 → [𝑦 / 𝑥] ¬ 𝜑) ↔ (𝑦 ∈ 𝐴 → ¬ [𝑦 / 𝑥]𝜑))
6	5	albii 1820	. . . 4 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → [𝑦 / 𝑥] ¬ 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐴 → ¬ [𝑦 / 𝑥]𝜑))
7		imnang 1842	. . . 4 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → ¬ [𝑦 / 𝑥]𝜑) ↔ ∀𝑦 ¬ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑))
8		alnex 1782	. . . 4 ⊢ (∀𝑦 ¬ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ ¬ ∃𝑦(𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑))
9	6, 7, 8	3bitri 299	. . 3 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → [𝑦 / 𝑥] ¬ 𝜑) ↔ ¬ ∃𝑦(𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑))
10	9	con2bii 360	. 2 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ ¬ ∀𝑦(𝑦 ∈ 𝐴 → [𝑦 / 𝑥] ¬ 𝜑))
11	3, 10	bitr4i 280	1 ⊢ (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535  ∃wex 1780  [wsb 2069   ∈ wcel 2114  ∀wl-ral 34833  ∃wl-rex 34834

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 1970  ax-7 2015  ax-10 2145  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070  df-wl-ral 34838  df-wl-rex 34848

This theorem is referenced by:  wl-dfreusb  34859