Theorem exlimiieq2 36837

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 36835	. 2 ⊢ ∃𝑦 𝑥 = 𝑦
4	1, 2, 3	exlimii 36833	1 ⊢ 𝜑

Syntax hints:   → wi 4  Ⅎwnf 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-12 2176  ax-13 2376

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-nf 1783

This theorem is referenced by: (None)