Theorem dfsb7ALT 2616

Description: Alternate version of dfsb7 2284. (Contributed by NM, 28-Jan-2004.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
dfsb1.p9	⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))

Assertion

Ref	Expression
dfsb7ALT	⊢ (𝜃 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑)))

Distinct variable groups:   𝑦,𝑧   𝜑,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜃(𝑥,𝑦,𝑧)

Proof of Theorem dfsb7ALT

Step	Hyp	Ref	Expression
1		dfsb1.p9	. 2 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
2		nfv 1914	. 2 ⊢ Ⅎ𝑧𝜑
3	1, 2	sb7fALT 2615	1 ⊢ (𝜃 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-10 2144  ax-11 2160  ax-12 2176  ax-13 2389

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069

This theorem is referenced by: (None)