Theorem exlimiieq1 37326

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

Hypotheses

Ref	Expression
exlimiieq1.1	⊢ Ⅎ𝑥𝜑
exlimiieq1.2	⊢ (𝑥 = 𝑦 → 𝜑)

Assertion

Ref	Expression
exlimiieq1	⊢ 𝜑

Proof of Theorem exlimiieq1

Step	Hyp	Ref	Expression
1		exlimiieq1.1	. 2 ⊢ Ⅎ𝑥𝜑
2		exlimiieq1.2	. 2 ⊢ (𝑥 = 𝑦 → 𝜑)
3		ax6e 2417	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
4	1, 2, 3	exlimii 37323	1 ⊢ 𝜑

Syntax hints:   → wi 4  Ⅎwnf 1806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-12 2215  ax-13 2406

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803  df-nf 1807

This theorem is referenced by: (None)