Theorem ss2abim 3994

Description: Class abstractions in a subclass relationship. Reverse direction of ss2ab 3995 which requires fewer axioms. (Contributed by SN, 22-Dec-2024.)

Assertion

Ref	Expression
ss2abim	⊢ (∀𝑥(𝜑 → 𝜓) → {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓})

Proof of Theorem ss2abim

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		spsbim 2084	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓))
2		df-clab 2720	. . 3 ⊢ (𝑡 ∈ {𝑥 ∣ 𝜑} ↔ [𝑡 / 𝑥]𝜑)
3		df-clab 2720	. . 3 ⊢ (𝑡 ∈ {𝑥 ∣ 𝜓} ↔ [𝑡 / 𝑥]𝜓)
4	1, 2, 3	3imtr4g 298	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝑡 ∈ {𝑥 ∣ 𝜑} → 𝑡 ∈ {𝑥 ∣ 𝜓}))
5	4	ssrdv 3923	1 ⊢ (∀𝑥(𝜑 → 𝜓) → {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓})

Syntax hints:   → wi 4  ∀wal 1546  [wsb 2074   ∈ wcel 2121  {cab 2719   ⊆ wss 3885

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  ax-6 1975  ax-7 2016

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-sb 2075  df-clab 2720  df-ss 3902

This theorem is referenced by:  ss2rabd  4006  moabex  5400