Theorem equs4 1653

Description: Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.)

Assertion

Ref	Expression
equs4	⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))

Proof of Theorem equs4

Step	Hyp	Ref	Expression
1		a9e 1626	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
2		19.29 1551	. . 3 ⊢ ((∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥 𝑥 = 𝑦) → ∃𝑥((𝑥 = 𝑦 → 𝜑) ∧ 𝑥 = 𝑦))
3	1, 2	mpan2 415	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥((𝑥 = 𝑦 → 𝜑) ∧ 𝑥 = 𝑦))
4		ancl 311	. . . 4 ⊢ ((𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → (𝑥 = 𝑦 ∧ 𝜑)))
5	4	imp 122	. . 3 ⊢ (((𝑥 = 𝑦 → 𝜑) ∧ 𝑥 = 𝑦) → (𝑥 = 𝑦 ∧ 𝜑))
6	5	eximi 1531	. 2 ⊢ (∃𝑥((𝑥 = 𝑦 → 𝜑) ∧ 𝑥 = 𝑦) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
7	3, 6	syl 14	1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))

Syntax hints:   → wi 4   ∧ wa 102  ∀wal 1282   = wceq 1284  ∃wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  sb2  1690  equs45f  1723