Theorem sb5fALT 2557

Description: Alternate version of sb5f 2492. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dfsb1.p5	⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
sb6fALT.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
sb5fALT	⊢ (𝜃 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))

Proof of Theorem sb5fALT

Step	Hyp	Ref	Expression
1		dfsb1.p5	. . 3 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
2		sb6fALT.1	. . 3 ⊢ Ⅎ𝑦𝜑
3	1, 2	sb6fALT 2556	. 2 ⊢ (𝜃 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
4	2	equs45f 2439	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
5	3, 4	bitr4i 279	1 ⊢ (𝜃 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1520  ∃wex 1761  Ⅎwnf 1765

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-10 2112  ax-12 2141  ax-13 2344

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-ex 1762  df-nf 1766

This theorem is referenced by:  sb7fALT  2570