Theorem bj-equsal1ti 37313

Description: Inference associated with bj-equsal1t 37312. (Contributed by BJ, 30-Sep-2018.)

Hypothesis

Ref	Expression
bj-equsal1ti.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
bj-equsal1ti	⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑)

Proof of Theorem bj-equsal1ti

Step	Hyp	Ref	Expression
1		bj-equsal1ti.1	. 2 ⊢ Ⅎ𝑥𝜑
2		bj-equsal1t 37312	. 2 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1559  Ⅎwnf 1804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-12 2213  ax-13 2404

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1801  df-nf 1805

This theorem is referenced by:  bj-equsal1  37314  bj-equsal2  37315