Theorem exlimiieq2 37727

Description: Inferring a theorem when it is implied by an equality which may be true. (Contributed by BJ, 15-Sep-2018.) (Revised by BJ, 30-Sep-2018.)

Hypotheses

Ref	Expression
exlimiieq2.1	⊢ Ⅎ𝑦𝜑
exlimiieq2.2	⊢ (𝑥 = 𝑦 → 𝜑)

Assertion

Ref	Expression
exlimiieq2	⊢ 𝜑

Proof of Theorem exlimiieq2

Step	Hyp	Ref	Expression
1		exlimiieq2.1	. 2 ⊢ Ⅎ𝑦𝜑
2		exlimiieq2.2	. 2 ⊢ (𝑥 = 𝑦 → 𝜑)
3		ax6er 37725	. 2 ⊢ ∃𝑦 𝑥 = 𝑦
4	1, 2, 3	exlimii 37723	1 ⊢ 𝜑

This proof depends on syntax axioms:   → wi 4  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2213  ax-13 2402

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817

This theorem is used by: (None)