Theorem iotabidv 4361

Description: Formula-building deduction rule for iota. (Contributed by NM, 20-Aug-2011.)

Hypothesis

Ref	Expression
iotabidv.1	⊢ (φ → (ψ ↔ χ))

Assertion

Ref	Expression
iotabidv	⊢ (φ → (℩xψ) = (℩xχ))

Distinct variable group:   φ,x

Allowed substitution hints:   ψ(x)   χ(x)

Proof of Theorem iotabidv

Step	Hyp	Ref	Expression
1		iotabidv.1	. . 3 ⊢ (φ → (ψ ↔ χ))
2	1	alrimiv 1631	. 2 ⊢ (φ → ∀x(ψ ↔ χ))
3		iotabi 4349	. 2 ⊢ (∀x(ψ ↔ χ) → (℩xψ) = (℩xχ))
4	2, 3	syl 15	1 ⊢ (φ → (℩xψ) = (℩xχ))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642  ℩cio 4338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621  df-uni 3893  df-iota 4340

This theorem is referenced by:  csbiotag  4372  ncfineq  4474  tfineq  4489  fveq1  5328  fveq2  5329  csbfv12g  5337  fvres  5343  tceq  6159