Theorem equs4 1718

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 1689	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
2		19.29 1613	. . 3 ⊢ ((∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥 𝑥 = 𝑦) → ∃𝑥((𝑥 = 𝑦 → 𝜑) ∧ 𝑥 = 𝑦))
3	1, 2	mpan2 423	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥((𝑥 = 𝑦 → 𝜑) ∧ 𝑥 = 𝑦))
4		ancl 316	. . . 4 ⊢ ((𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → (𝑥 = 𝑦 ∧ 𝜑)))
5	4	imp 123	. . 3 ⊢ (((𝑥 = 𝑦 → 𝜑) ∧ 𝑥 = 𝑦) → (𝑥 = 𝑦 ∧ 𝜑))
6	5	eximi 1593	. 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 103  ∀wal 1346   = wceq 1348  ∃wex 1485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  sb2  1760  equs45f  1795