Theorem bj-spimtv 32718

Description: Version of spimt 2253 with a dv condition, which does not require ax-13 2246. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-spimtv	⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem bj-spimtv

Step	Hyp	Ref	Expression
1		ax6ev 1890	. . . 4 ⊢ ∃𝑥 𝑥 = 𝑦
2		exim 1761	. . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥(𝜑 → 𝜓)))
3	1, 2	mpi 20	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ∃𝑥(𝜑 → 𝜓))
4		19.35 1805	. . 3 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
5	3, 4	sylib 208	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → ∃𝑥𝜓))
6		19.9t 2071	. . 3 ⊢ (Ⅎ𝑥𝜓 → (∃𝑥𝜓 ↔ 𝜓))
7	6	biimpd 219	. 2 ⊢ (Ⅎ𝑥𝜓 → (∃𝑥𝜓 → 𝜓))
8	5, 7	sylan9r 690	1 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1481  ∃wex 1704  Ⅎwnf 1708

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-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)