Theorem sbequiALT 2596

Description: Alternate version of sbequi 2091. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 15-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dfsb1.xz	⊢ (𝜃 ↔ ((𝑧 = 𝑥 → 𝜑) ∧ ∃𝑧(𝑧 = 𝑥 ∧ 𝜑)))
dfsb1.yz	⊢ (𝜏 ↔ ((𝑧 = 𝑦 → 𝜑) ∧ ∃𝑧(𝑧 = 𝑦 ∧ 𝜑)))

Assertion

Ref	Expression
sbequiALT	⊢ (𝑥 = 𝑦 → (𝜃 → 𝜏))

Proof of Theorem sbequiALT

Step	Hyp	Ref	Expression
1		equtr 2028	. . 3 ⊢ (𝑧 = 𝑥 → (𝑥 = 𝑦 → 𝑧 = 𝑦))
2		dfsb1.xz	. . . . 5 ⊢ (𝜃 ↔ ((𝑧 = 𝑥 → 𝜑) ∧ ∃𝑧(𝑧 = 𝑥 ∧ 𝜑)))
3	2	sbequ2ALT 2580	. . . 4 ⊢ (𝑧 = 𝑥 → (𝜃 → 𝜑))
4		dfsb1.yz	. . . . 5 ⊢ (𝜏 ↔ ((𝑧 = 𝑦 → 𝜑) ∧ ∃𝑧(𝑧 = 𝑦 ∧ 𝜑)))
5	4	sbequ1ALT 2579	. . . 4 ⊢ (𝑧 = 𝑦 → (𝜑 → 𝜏))
6	3, 5	syl9 77	. . 3 ⊢ (𝑧 = 𝑥 → (𝑧 = 𝑦 → (𝜃 → 𝜏)))
7	1, 6	syld 47	. 2 ⊢ (𝑧 = 𝑥 → (𝑥 = 𝑦 → (𝜃 → 𝜏)))
8		ax13 2393	. . 3 ⊢ (¬ 𝑧 = 𝑥 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
9		sp 2182	. . . . . 6 ⊢ (∀𝑧 𝑧 = 𝑥 → 𝑧 = 𝑥)
10	9	con3i 157	. . . . 5 ⊢ (¬ 𝑧 = 𝑥 → ¬ ∀𝑧 𝑧 = 𝑥)
11	2	sb4ALT 2588	. . . . 5 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (𝜃 → ∀𝑧(𝑧 = 𝑥 → 𝜑)))
12	10, 11	syl 17	. . . 4 ⊢ (¬ 𝑧 = 𝑥 → (𝜃 → ∀𝑧(𝑧 = 𝑥 → 𝜑)))
13		equeuclr 2030	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑦 → 𝑧 = 𝑥))
14	13	imim1d 82	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑧 = 𝑥 → 𝜑) → (𝑧 = 𝑦 → 𝜑)))
15	14	al2imi 1816	. . . . 5 ⊢ (∀𝑧 𝑥 = 𝑦 → (∀𝑧(𝑧 = 𝑥 → 𝜑) → ∀𝑧(𝑧 = 𝑦 → 𝜑)))
16	4	sb2ALT 2587	. . . . 5 ⊢ (∀𝑧(𝑧 = 𝑦 → 𝜑) → 𝜏)
17	15, 16	syl6 35	. . . 4 ⊢ (∀𝑧 𝑥 = 𝑦 → (∀𝑧(𝑧 = 𝑥 → 𝜑) → 𝜏))
18	12, 17	syl9 77	. . 3 ⊢ (¬ 𝑧 = 𝑥 → (∀𝑧 𝑥 = 𝑦 → (𝜃 → 𝜏)))
19	8, 18	syld 47	. 2 ⊢ (¬ 𝑧 = 𝑥 → (𝑥 = 𝑦 → (𝜃 → 𝜏)))
20	7, 19	pm2.61i 184	1 ⊢ (𝑥 = 𝑦 → (𝜃 → 𝜏))

Syntax hints:  ¬ wn 3   → 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:  sbequALT  2597