Theorem bj-spimtv 36795

Description: Version of spimt 2391 with a disjoint variable condition, which does not require ax-13 2377. (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 1969	. . . 4 ⊢ ∃𝑥 𝑥 = 𝑦
2		exim 1834	. . . 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 1877	. . 3 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
5	3, 4	sylib 218	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → ∃𝑥𝜓))
6		19.9t 2204	. . 3 ⊢ (Ⅎ𝑥𝜓 → (∃𝑥𝜓 ↔ 𝜓))
7	6	biimpd 229	. 2 ⊢ (Ⅎ𝑥𝜓 → (∃𝑥𝜓 → 𝜓))
8	5, 7	sylan9r 508	1 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538  ∃wex 1779  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-12 2177

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784

This theorem is referenced by: (None)