Theorem bj-substw 34641

Description: Weak form of the LHS of bj-substax12 34640 proved from the core axiom schemes. Compare ax12w 2133. (Contributed by BJ, 26-May-2024.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-substw.is	⊢ (𝑥 = 𝑡 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
bj-substw	⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑))

Distinct variable group:   𝜓,𝑥

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

Proof of Theorem bj-substw

Step	Hyp	Ref	Expression
1		bj-substw.is	. . . . 5 ⊢ (𝑥 = 𝑡 → (𝜑 ↔ 𝜓))
2	1	pm5.32i 578	. . . 4 ⊢ ((𝑥 = 𝑡 ∧ 𝜑) ↔ (𝑥 = 𝑡 ∧ 𝜓))
3	2	exbii 1855	. . 3 ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝑡 ∧ 𝜓))
4		19.41v 1958	. . 3 ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜓) ↔ (∃𝑥 𝑥 = 𝑡 ∧ 𝜓))
5	3, 4	bitri 278	. 2 ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) ↔ (∃𝑥 𝑥 = 𝑡 ∧ 𝜓))
6	1	biimprcd 253	. . 3 ⊢ (𝜓 → (𝑥 = 𝑡 → 𝜑))
7	6	alrimiv 1935	. 2 ⊢ (𝜓 → ∀𝑥(𝑥 = 𝑡 → 𝜑))
8	5, 7	simplbiim 508	1 ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1541  ∃wex 1787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788

This theorem is referenced by: (None)