Theorem sb4aALT 2574

Description: Alternate version of sb4a 2498. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
dfsb1.p2	⊢ (𝜃 ↔ ((𝑥 = 𝑦 → ∀𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑)))

Assertion

Ref	Expression
sb4aALT	⊢ (𝜃 → ∀𝑥(𝑥 = 𝑦 → 𝜑))

Proof of Theorem sb4aALT

Step	Hyp	Ref	Expression
1		dfsb1.p2	. . 3 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → ∀𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑)))
2	1	sb1ALT 2558	. 2 ⊢ (𝜃 → ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑))
3		equs5a 2469	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
4	2, 3	syl 17	1 ⊢ (𝜃 → ∀𝑥(𝑥 = 𝑦 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-12 2175  ax-13 2379

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786

This theorem is referenced by:  sb6fALT  2578