Theorem equs4v 1930

Description: Version of equs4 2290 with a dv condition, which requires fewer axioms. (Contributed by BJ, 31-May-2019.)

Assertion

Ref	Expression
equs4v	⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem equs4v

Step	Hyp	Ref	Expression
1		ax6ev 1890	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		exintr 1819	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
3	1, 2	mpi 20	1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))

Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1481  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-6 1888

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  equvelv  1963  bj-sb56  32639  bj-equs45fv  32752  bj-sb2v  32753