Theorem dfsb7OLD 2286

Description: Obsolete version of dfsb7 2285 as of 3-Sep-2023. (Contributed by NM, 28-Jan-2004.) Revise df-sb 2070. (Revised by BJ, 25-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
dfsb7OLD	⊢ ([𝑡 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))

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

Allowed substitution hints:   𝜑(𝑥,𝑡)

Proof of Theorem dfsb7OLD

Step	Hyp	Ref	Expression
1		df-sb 2070	. 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		sb56 2277	. 2 ⊢ (∃𝑦(𝑦 = 𝑡 ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3		sb56 2277	. . . . 5 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
4	3	bicomi 226	. . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
5	4	anbi2i 624	. . 3 ⊢ ((𝑦 = 𝑡 ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ (𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
6	5	exbii 1848	. 2 ⊢ (∃𝑦(𝑦 = 𝑡 ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
7	1, 2, 6	3bitr2i 301	1 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535  ∃wex 1780  [wsb 2069

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

This theorem is referenced by: (None)