Theorem sb4aALT 2598

Description: Alternate version of sb4a 2509. (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 2582	. 2 ⊢ (𝜃 → ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑))
3		equs5a 2480	. 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 208   ∧ wa 398  ∀wal 1535  ∃wex 1780

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  ax-13 2390

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785

This theorem is referenced by:  sb6fALT  2602