Theorem spimeh 1667

Description: Existential introduction, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by NM, 3-Feb-2015.) (New usage is discouraged.)

Hypotheses

Ref	Expression
spimeh.1	⊢ (𝜑 → ∀𝑥𝜑)
spimeh.2	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
spimeh	⊢ (𝜑 → ∃𝑥𝜓)

Proof of Theorem spimeh

Step	Hyp	Ref	Expression
1		a9e 1626	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		spimeh.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3		spimeh.2	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	3	com12 30	. . 3 ⊢ (𝜑 → (𝑥 = 𝑦 → 𝜓))
5	2, 4	eximdh 1542	. 2 ⊢ (𝜑 → (∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜓))
6	1, 5	mpi 15	1 ⊢ (𝜑 → ∃𝑥𝜓)

Syntax hints:   → wi 4  ∀wal 1282   = wceq 1284  ∃wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115

This theorem is referenced by: (None)