Theorem bnj133 30793

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj133.1	⊢ (𝜑 ↔ ∃𝑥𝜓)
bnj133.2	⊢ (𝜒 ↔ 𝜓)

Assertion

Ref	Expression
bnj133	⊢ (𝜑 ↔ ∃𝑥𝜒)

Proof of Theorem bnj133

Step	Hyp	Ref	Expression
1		bnj133.1	. 2 ⊢ (𝜑 ↔ ∃𝑥𝜓)
2		bnj133.2	. . 3 ⊢ (𝜒 ↔ 𝜓)
3	2	exbii 1774	. 2 ⊢ (∃𝑥𝜒 ↔ ∃𝑥𝜓)
4	1, 3	bitr4i 267	1 ⊢ (𝜑 ↔ ∃𝑥𝜒)

Syntax hints:   ↔ wb 196  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by:  bnj150  30946  bnj983  31021  bnj984  31022  bnj985  31023  bnj1090  31047  bnj1514  31131