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Theorem bj-ax12ig 34043
 Description: A lemma used to prove a weak form of the axiom of substitution. A generalization of bj-ax12i 34044. (Contributed by BJ, 19-Dec-2020.)
Hypotheses
Ref Expression
bj-ax12ig.1 (𝜑 → (𝜓𝜒))
bj-ax12ig.2 (𝜑 → (𝜒 → ∀𝑥𝜒))
Assertion
Ref Expression
bj-ax12ig (𝜑 → (𝜓 → ∀𝑥(𝜑𝜓)))

Proof of Theorem bj-ax12ig
StepHypRef Expression
1 bj-ax12ig.1 . . . 4 (𝜑 → (𝜓𝜒))
21pm5.32i 578 . . 3 ((𝜑𝜓) ↔ (𝜑𝜒))
3 bj-ax12ig.2 . . . . 5 (𝜑 → (𝜒 → ∀𝑥𝜒))
43imp 410 . . . 4 ((𝜑𝜒) → ∀𝑥𝜒)
51biimprcd 253 . . . 4 (𝜒 → (𝜑𝜓))
64, 5sylg 1824 . . 3 ((𝜑𝜒) → ∀𝑥(𝜑𝜓))
72, 6sylbi 220 . 2 ((𝜑𝜓) → ∀𝑥(𝜑𝜓))
87ex 416 1 (𝜑 → (𝜓 → ∀𝑥(𝜑𝜓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811 This theorem depends on definitions:  df-bi 210  df-an 400 This theorem is referenced by:  bj-ax12i  34044
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